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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http : //books . google . com/ MATHEMATICS. COMPILED FROM THE BEST AUTHORS AMD IXTBNOEO TO BE THB TEXT-BOOK OF THE CDUts^c of ^xiMtt %tttnxti ON THESE SCIENCES IK THI University at Cambridge. UNDER THE DIRECTION OF SAMUEL WEBBER, ji.m.ji.a.s, HOLLI8 PROFESSOR OF MATHEMATICS AND NATVRAL PSlLOtOPflr. IN TWO VOLUMES. — VOL. I. C0P2 li^iff&t SZtUXZtt. PRINTED FOR THE UNirERSITT AT CAMBS'IDGJ^ BY THOMAS i5f ANDREWS. iSoi, # Klf^rr /^ \^di/c%iuemc7i6. 1 HE design, in making this Com- pilation, is to collect suitable ex^cises to be performed by .the Classes at the Private Lec- tures on Mathematics, given in the Univer- sity. The view, which the Corporation had, of the great advantages, that the Students might derive from a judicious work of the Idnd, produced this attempt to promote their improvement The parts of the most ap- proved writings, selected for the purpose, are copied, with only such alterations, as appeared to be useful The Authors of most of the branches are Dr. Hutton and Mr. Bonny- castle J the Navigation is principally from that of Mr. Nicholson ; and considerable use has been made of a Manuscript^ Mole's Algebra^ and the Works of Emerson. Cotttenttf OF THE fixat WAnmt. ARITHMETIC I Explanation or characters . » « • 9 Notatioe • • • - • • .' • • • 'it Simple Additioo . • • . . • ' • • 14 Subtraction ' - • ^ • % • • . t9 Moltiplicatioii *.*•••• 96 ^iTi^on •*• •'• «*• • • M RedactioQ * . - « ^ « » • ' • • • • 36 CoiDpound Additionr . • • • • • • 4* Subtraction . ^ . • * • . • ^ 46 Mtdtiplication . - • * • • • • 50 DivisioD •' • • « • • • 53 DUODUIXMALS ••.••••• 5i» Vulgar Fractioks •••••• 59 Reduction of Vuigar Ftactions • • . • • 63 Addition of do. 72 Subtraction of do 74 Multiplication of do • ibid Division of do. . . * • • • • 75 Decimal Fractions 76 Addition of* Dedmais • * 78 Subtracticm of do. • • ^ « • • • ibid Multiplication of do. « • » . • • -79 Division of do 82 Reduction of do^ . ,. 85 J'ederal CONTENTS. FEDERAt M^NEir Keducuon of Circolatiog Decimals Addition of do. jSubtractioo of do. • • • Multiplication of do* Division of do. • • • « PftOPORTtOW' Xtf GENlfll4t» • fcmple Proportion, or Rule of Three F^tice • • • ^ • "JCarc and Trett Convpo^nd Propor^oik Cpojoined do* Btrter Jjyss and Gain • £ingle Fellowship Double do. . » Alligation Medial Alternate Ittvolution •. Evolution . • f Extraction 'of die Squire Root Cube Root '- Cube Root by ApproKimatton Roots -of Powers in genei^ Do. by Approximation , Arithmetical -PrognessioD- . • . . , Geometrical da. *. ". . , . Simple Interest . . , , by Decimals . . . Commission ^ .. , , ^ « Brokerage • • . ft » Insurance , . . . . ..... .Discount .. .. .. .. .. .. f by Decimals .... * / Page 90 93 94 99 100 ibid lOI loa 105 M» "5 129 i3« 133 iSf «37 140 14a 150 158 160 165 166 16S i8x 185 186 18: i8{ i8f I' Equat COMTEMTS. TU '« faff E^ltttfioii of Biyments ...» I • 19; ky Decunals • • • • 197 CompoBttd Interest « tot by Decimals » • • • S04 Aaftoittes at Simple Interest S07 Compound do. • • • • • sio VaKie of ao Annuity for ever, at Cbmpoond Interest • • ttg Value of an Annuity in Reversiont at Conqpound Interest tiS Single Position . • • • • • • J it- Double do, ••,••••• aao Permuution, Combination and Composition of Qoantittes t%$ Miscellaneous Questions • 2$J LOGARITHMS- LOGA&ITHMS • . •» « • • » • S43 Compaution of Logarithnjs • • • • • 346 Description and Use of the Table of Logarithms • • 250 Multiplication by Logarithms 255 Division by do. ••••••• 257 Involution by do. • • • . • • 258 Evolution by do. 260 • ALGEBRA. Definitions and Notation 265 Addition . * 270 Subtraction 275 Multiplication -277 Division ••••«.••• 287 Fractions 294 Involution 3^^ Evolution . . • • . . • • -3** Sards • 3^<^ Infinite Series . . • • • • • • 3^^ f^'rriplc viU * €ONTEKT«r ' Simple* Equations • • - • • • • • ^ 53 j Reduction of two, tliree, or more Simple Equations to gw 339 Quadratic Eruattons .-••..• • « 354. Cubic and Higher Equations • • • . . 370 * • ■ GEOMETRY. Definitions • • ... • • • • «38t Problems • • ,\ » * r < • • 39a* Construction of tlie Plane Scale (Prolx L*) • • . 42b EXPLANATION of CHARACTERS. SIGNIFIES equality : as 20 shillings rr I pound, sig- nifiesy that 20 shillings are equal to one pouuiL -j- Signifies plus^ or addition : as, 4 -f- ^ ^^ 4 — Signifies minus^ or subtraction : as, 6— 211:4. X IntOy signifies multipiication : as, 3 X 2 rz 6. ^ JSy, or ) { signifies division : as, 6-7-2 = 3, or 2)6(3. Division may alfo be denoted by placing the dividend over a line,'and the divisor under it: thus4'=:6 -f- 2^:3. **::•• Signifies arithmetical proportion : thus 2" 4 : : 6 •• 8} here the meaning is, that 4 — 2=:8 — 6=2. ; : : : Signifies geometrical proportion : thus 2:4:13:6, which is to be read, as 2 to 4, fo is 3 to 6. •T" Signifies arithmetical progression* -^ Signifies continual geometrical proportion^ or geometrical progression. *•, Signifies therefore* — ^1* Signifies the second power^ or squan. — 1^ Signifies the third power j or cube. **ni'" Signifies any poiver. ^, or""jJ, Signifies the square root. B lO EXrLA>JATlON OF CHARACTERS. 3^, or I', Signifies the cube root, s/y 01^ j". Signifies m^y root. m ' p Signifies any root of any power. Note. Tht number, or letter, belonging fo the aboTie s%ns^ of powers and roots, is called the indexy or exponent. A line, or vinculum, drawn over several numbers, sig- nifies, that file numbers under it are to-be considered jointly : thus, 2o — 7-{-S^=5> but without die vincu^ lum, 20 — 74-8=21. cbzzay^b z:z the product of a and k ARITHMETIC. >*«v^'v^(^hs^n(>(V)^i^ JljLRITHMETIC is the art of computing by numbers, and has five principal or fundamental rules for its opera- tions ; viz. Notation^ Addition, Subtraction, Muhiplica^ tion, ^nd Division. NOTATION* Notation teacheth how to express any proposed number, cither by words or characters. To * As it is absolutely necessary to have a perfeft knowledge of our excellent method of notation, in order to understand' the reasoning made use of in the following notes, I shall endeavour to explain it in as clear and concise a manner as possible. First) then, it may be observed, that the characters, by which all numbers are expressed, are these ten ; o, i, 2, 3, 4, 5, 6, 7, 8, 9 ; o is called a cypher^ and tlie rest, or rather all of them, are -called figttres or dlgtis. The names and signification of tliese characters, and the origin or generation of the numbers they stand for, are as follow : o nothing ; i one, or a single thing called aa Ainit ; .1 + 1=2 two; 2+1=3 three; 3 + 1=4 four; 4+1 = ^ five; 5+1=6 six; 6+1 = 7 feven ; 7+1 = 8 eight ; 8 + 1=9 .nine ; and 9+i=ten; which has no single character ; and thus by the continual ad4ition of pnc, all numbers are generated. I. Beside tZ ARITHMETIC. Thread NUMBERS. To the simple value, of each figure join the name of Its place, beginning at the left hand and reading toward the right. EXAMPLES. Read the following numbers : 37 30791 iiioooiii loi 70079 1234567890 1107 3306677 J0203Q405060708090. r^ 2. Beside the simple value of the figures, as above noted, they have, each, a local value, according to the following law : Viz. In a combination of, figures, reckoning from right to left, the figure in the first place represents its primitive simple value ; that itt the second place, ten times its simple value ; that in the tliird place, a hundred times its simple value ; and so on ;^the value of the figure in each succeeding place being ten times the value of it in that immediately preceding it. 3. The names of the places are denominated according to their order. The first is called the place of units ; the second, tens ; the third, hundreds ; the fourth, thousands ; the fifth, ten thou- sands ; tlie sixth, hundred thousands ; the seventh, millions ; and so on. Thus in the number 3456789 ; 9 in the first place signi- fies only nine ; 8 in the second place signifies eight tens, or eighty; 7 in the third place is seven hundred ; 6 in the fourth place is six thousand ; 5 in the fifth place is fifty thousand ; 4 in the sixth place is four hundred thousand ; and 3 in the seventh place is three millions ; and the whole number is read thus, three millions, four hundred and fifty six thousand, spvcn hundred and eighty nine. 4. A cypher, though it signifies nothing of itself, yet it occupies a place, and, when set on the right hand of other figures, in- creases their value ,in the same ten-fold proportion ; thus, 5 sig- nifies only i^:^^y but 50 is five ten^ or fifty, . and 500 is 9is^ hundred, &c. r. Foi notation; ^^ Ta nvrite NUMBERS. \, RULE. \^ Write down the figures in the same order their Taloei are expressed in, beginning at the left hand, and writing toward the right ; remembering to supply those places of the natural order with cyphers, which are omitted in the question^ EXAMPLES, 9 •> 5. For the xnpre ea^y reading of large numbers^ they arc <£- Tided into periods, and half periods, each half period consisting of three figures ; the name of the £rst period being units ; cf the second, millions ; of the third, billions ; of the fourth, trilljons» &c. Also the first part of any period is so many units of it^ and the latter part, so many thousands. The JbUowlng Talk eotUains a summary of the mfhole doSrimk Iftiwdi. (^adrfl. TriiTI BOlions. MiUIoM. Units, f ^k^^^nsJ K^^/-\J UorO Vxw; U'w^l Half Per. th. nn. th. un. th. on. th. nn. cxt cxn I ^V^^vO VjlTLfl-rinJ ^_ |-y -^_f^_ r^- ^J ^_|-^-^J^_|-^-^J ^■^- j^-^-^j 1 Figores. iz^,4?6 789,098 7g5,43x 101,234 ^6-^.Hoo | ji Synopsis of the Roman Notjtiow* 2=zII As often as any character Is repeated, so many 3=1X1 times is its value repeated. 4=:IIII or IV A less character before a greater dimin- 5=V ishes its value. 6=VI A less character after a greater increases its 7=VII value. 8=VIII 9=IX jo=X 50=1. ioo=C 500=0 or ID For every afSxed, this becomes lo times as many. looo / EXAMPLES. ^ Write down in figures the following numbers ; £ighty one. Two hundred and eleven. One^HHisanj jmd thirty nine. A million jmd a half. A hundred an^^ four-score and £ve thousand* Eleven thousand miUion^ eleven hundred thousand and .eleven. Thirteen billionj ^\x hundred thousand million^ fouf tho^isand and one. SIMPLE ADDITION. Zimph Addition teacheth to collect $cvpi^ num'bers of •Ifhe same denomination into one totaL RULE.* I. Place the numbers under each other, 30 that units maf stand under uzi^ts, ten$ under tigns^ &c. and draw a , line under them. 2. Add looorrM or CID For every C and 3, put one at each end, 20OO=MM it becomes ten times as much. 5000=103 ; orV A line over any number increases 6ooo=VI ^ *ooo foM. ioooo=X or CCIOO 50000=1333 60000= LJC iooooo=C or CCCI333 % ioooooo=M or CCCCI3333 2 000000= MM &c. &c. ^ This rule, as well as the method of proof, is founded oa the known axiom, " the wholo is equal to the sun) of all its parts.'' AH that requires explaining is the method of placing the num- bers. iiuvtt ADDirroii. tj di Add up the figures in the row of units, and find how inany tens are contained in their sum. 3. Set down the reqiainder, and carry as many tinits to the next row, as there are tens 5 with which prdceed as before ; and so on till the whole k finished. Method fcers, and carrying for the tens ; both which are evident firom the ttatare of notation : for any other disposition of the numbers womki entirely alter their value ; and carrying one for every teD» from an inferior line to a sv^erior^ is evidently right, since an unit IB the latter case is of the fas>e value as ten in the former; Beside the method here given, therie is another very ingenious one of proving addition by casting out the nines, thus : . RiTLB I. Add the figures ia the uppermost line together, aii4 £ad bow Biany nines are contaii^ in tlieir sum. 2. Reject the, nine$, and set down the remainder directly erett with the figures in the line* J. Do the same with each of the given numbers, and- set all these excesses of nine together in a row, and find their sum ^ then if the excess of nines ib this sum, found as before, is equal \a the excess of nines in the total sum^ the question is ri^t* -EXAM PLE. 3782 A 38 5766 •16 8755 *&.? 1>— ■ 18303 4^ This method depends upon a property of the number 9, whicfi belongs to no other digit whatever, except 3 ; viz. that any num- ber, divided by 9, will leave the sa;ne remainder as the sum of its figures or dibits divided by 9 ; which may be thus demonstrated. Demon* itf AttlTHMETia ' . Method of PrOQF. I. Draw a line below the uppermost number, and sup- pose it cut off. 2* Add all the rest together, and set their sum under iSac number to be proved* 3'. Add Demon. Let there be any nombery as 34J67 ;. this separated into its several parts becomes 3000+400+60-1-7 ; but 3000=3 Xlooc==3X 999 +1=3X999 +3* In like manner 4002=4 X 99+4, and 60=6x9+^- Therefore 34^7=3 X999+3+4X 99+4+6X9+6+7=3x999+4x99+6x9+3+4+6+7. And3467^ 3X999+4X99+6X9 3+4+6+7» 3^^ ^^ ^ 9 9 .. ^ 909+4X99+6x9 is evidently divisible by 9 ; therefoie 3467 divided by 9 will leave the same remainder as 3+4+6+7 di* vided by 9 ; and the same will hold for any other number what* ever. C^E* D. * The same may be demonstrated universally thus : Demon. Let Nzz any number whatever, a^ by Cy &:c. the digits of which it is composed, and«= as many cyphers as ay the highest digit, is places 'from imity, . Then Nz=a with », o's+3 with n — i,o*s+rwithii — 2, o's, &c. by the nature of notation ; =flX»— 1> 9's+tf+^X«— 2, 9's+^+rX«— 3, 9's + r, 3cc =flX«— i,9's+^X»--2,9's+^X»— 3, .9's, &c. +a+5+r, &c. butflX«— 1> 9*s +1 X «— 2, 9's +r X «— 3, 9's, &c. is plain- ly divisible by 9 ; therefore N divided by 9 will leave the same remainder, as fl+^+c, &c. divided by 9. Q^E. D. In the very same manner, this property may be shown to belong to the number three ; but the preference is usually given to the Bomber 9, on account of its being more convenient in practice. Now from the demonstration here given, the reason of the rule itself is evident ; for the excess of nines in two or more numbers bcmg taken separately, and the excess of nines taken also out of the ^ SIMPLE ADDITIOK. n 3. Add this last found number and the uppermost line together, and if their sum be the same as that found by the first addition^ the sum is right. EXAMPLES. 22345 (3) 34578 78961 23456 78901 2345<J 78901 67890 8752 340 350 78 3750 87 328 17 327 307071 Sum. 99755 Sum. 39087 Sum. ^83615 774IO- 4509 3^07071 Proof. 99755 Proof. 39087 Proof. 4. Add 8635, ai94, 742i> 5063, 2196, and 1245 to- gether. Alls. 26754. c. Add the sum of the former excesses, it is plain this last excess must be equal to the excess of nines contained in the total sum of all these numbers ; the parts being ttjual to the whole. This rule was first given by Dr. Wallis, in his Arithmetic^ published A. D. 1657, and is a very simple easy method ; though it is liable to this inconvenience, that a wrong operation may some- times appear to be right ; for if we change the places of any two figured in the sum, it will still be the same ; but then a true sum will always appear to be true by this proof ; and to make a false one appear true, there must be at least two errors, and these op- posite to each other ; and if there be more than two errors, they must balance among themselves : but the chance against this particular circumstance Is so great, that we may pretty safely tnwt to this proof. C iS^ 'iilTHMETIC. 5. Add 246034, 298765, 473^1, 58653, 64218, 5376; '9821, and 340 together. Ans. 7305a8« 6. Add 562163, 21964, 56321, 18536, 4340, 279, and 83 together. Ans. 663686* 7. How many shillings are there In a crown, a guinea, a nwidore, and a six and thirty ? Ans. 8g. 8. How m^y days are there in the twelve calendar months ? . Ans. 365. 9. How many days are there from the r9th day of April, 1774, to the 27th day ©f November, 1775, both days exclu- sive ? . Ans. 586; . SIMPLE SUBTRACtlOlSIv Simple Suki^aBion teacheth to take a less number /Mtt a greater of the same denomination, and thereby* sh^^ the difierence or remiatinden: The less number, or tJiat which is to be subtracted, is" called the subtrahend ; the other, the minuend: and the number diat is found by the operation, the remainder or difference. RULE.^ I. Place the less number under the greater, so tfcat units' may stand under units, teiis under tens, &c« and^draw a' line under them^ • 2. Begii* * Demon, i. When all the figures of the less numBcr arc ^ess than their correspondent figures in tke greater, ^e <Ufi*ereDce of the figures in the several like places must altogether make the true difference sought ; because as the sum of the parts is equal 10 the whole, so must the sum of the differences of all the similar parts be equal to the difference of the whole* 2. Whea SIMPLE SUBTRACTION. ^9 ' 2. Begin at the right hand, and take each figure in the lower line from the figure above it, and set down the rc.- piainder. ~ 3. If the lower figure is greater than that above it, add ten to the upper figure ; from which figure, so increased, take the lower, and set down the remainder^ carrying one to the next lower figure j with which proceed as before, and so on till the whole is finished. Method of Proof. Add the remainder to the less number^ and if the suio js equal t:o the greater, the work is right. (0 Fronj 32876I5 Take 2343756 EXAMPLES. Ffom 53274<>.7 Take 1008438 (3) From 1234567 Take 345673 Jlemaiild. 9438^^ Remain. 4319029 Remain. 8888^^9 proof 3287625 Proof 5327467 Proof 1234567 4. From 2. When any figure of the greater number is less than its cor- tcspondent figure in the less, the ten, which is added by the rule, is the value of an unit in the next higher place, by the nature of potation ; and the one that is added to the next place of the less number is to diminish the co-respondent place of the greater ac- cordingly ; which is only taking from one place and adding as much to another, "syhereby the total is never changed. And by this means the greater number is resolved into such parts, as are each greater than, or equal to, the similar parts of the less : and the difference of the corresponding figures, taken together, will evidently make up the difference of the whole. Q^E. D. The truth of the method of proof is evident : for the difference of two numbers, added to 'the less, is manifcftly equal to the greater. 30 ARITHMETIC. 4. From 2637804 take 2^j6^i2. • Ans. 260822. 5. From 3762162 take 826541. Ans. 2935621. 64 From 78213606 take 27821890. Ans. 50391716. 7. The Arabian method of notation was first known in England about the year 1150: how long was it thence to the year 1776 ? Ans, 626 years. 8. Sir Isaac Newton was born in the year 1^42, and 4ied in 1727 : how old was he at the time of his decease ? Ans. 85 years. SIMPLE MULTIPLICATION, Simple Multiplication is a compendious method of ad- dition, and teacheth to find the amount of any given number of one denomination, by repeating it any propos- ed number of times. The number to be multiplied is called the multiplicand. The number you multiply by is called the multiplier. The number found from the operation is called the proJucI, Both the multiplier and multiplicand are, in general, called terms or factors. ^ MULTIPLICATION AND plVISION TABLE. 1| 2 3| 4 5(6 7 1 S 1 9 1 10 11 [ \% M 4 $1 & io| la 14 M6 1 \%\ au> %%\ %^ ,11 6 9 1 "Mil ^^ ii \ M 1 57 1 30 33 1 3^ 4l » h \ 11 11 1 16 j zo 1 14 15 j *<J ^5 1 JO iS 1 3^ ( 36 1 40 44 1 4* 35 1 40 1 45 1 jo 55 1 60 4^|4S1 54! 6&f 'i^l 7* 7 1 14 f 21 %t 35 1 ^% 49 1 56 1 63 1 70 77 »4 H 1 16 1 %^ 3i 40 i 4^ 56 1 64 1 72 1 Sol S8 96 Q 1 18 1 17 56 1 45 1 54 63 f 72 1 Sr 1 90 99 1 log io 1 iO 1 30 40 1 ^^0 1 6d 70 j go [ 90 1 100 lio 1 1^0 "1 1 11 1 33 1 44 1 J5 1 66 \ 77 1 lis 1 j.9 1 110 1 lai 1 133 ,. I td 1 ,^6 (48 t Co 1 7a 84 \ 96 r^s 1 J 20 1 13a 1 144 J7i SIMPLE MULTIPLICATION. 2t I USJE of the Table in MULTIPLICATION. Find the multiplier in the left-hand column, and the multiplicand in the uppermost line 5 and the product is in the common angle of meeting, or against the multiplier, and under the multiplicand. Use of tie Table in Din 81027, Find the divisor in the left-hand column, .and the divi- dend in the same line ; then the quotient will bcovcrths divideud, at the top of the column. . • RULE.* \ I. Place the multiplier under the multiplicand, §0 that units may stand under units, tens under tens, ^c. and draw a line under them. 2. Begin * Demon, i. When the multiplier is a single digit, it is plain tkat we find the product ; for by multiplying every figure, that is, every part of the multiplicand, we multiply the whole ; and writing down the products that are less than len, or the excess of tens, in the places of the figures multiplied, and carrying the num^ ber of tens to the product of the next place, is only gathering to- gether the similar parts of the respective products, and is, there- fore, the same thing, in effect, as though we wrote down the multiplicand as often as the multiplier expresses, and added them together : for the sum of every cdlumn is the product of the figures in the place of that column ; and these products, collect- ed together, are evidently equal to the whole required product. Z. If the multiplier is a number made up of more than one digit. After we have found the product of the multiplicand by the first figure of the multiplier, as above, we suppose the multi- plier divided intd parts, and find, after the same manner, the pro- duct of the multiplicand by the second figure of the multiplier ; but as the figure we arc multiplying by stands in the p!;icc of tens ; ^:| AR]fTHMETlCt 2. Begin at the right hand, and multiply the whde m\»l-^ tiplicand sererally by each figure in the multiplier, setting dawn the first figure of every line diyectly under the fig- ure lens ; the ]>roduct mast be ten times its simple value ; and therey fore the first figure of this produa mast be placed in the* place of tens ; or, which is the same thing, .directly under the fig- ure we are multi|)Iying by. And proceeding in this manner sepa* rately yrith all the figures of 0ie multiplier, it is evident that we shall multiply all the parts of the multiplicand by all the parts of the multiplier ; or the wjiole of the multi^icand by the whole of the multiplier ; therefore these several products being added to- gether will be equal to the whole required product. Q^E. D. The reason of the method of proof depends upon this propo- sition, " that if two numbers are to be multiplied together, either of them may be made the multiplier, or the multiplicand, and the product will be the same." A small attention to the nature of numbers will make this truth evident : £>r 3 x 7=:2i3s7 X 3 ; and in general 3X4X5X6, &c. ==4x3x6x5* See. without any regard to the order of the terms : and this is true of any number of factors whatever. The following examples are subjoined to make the reason of the role appear as plain as possible. (0 37565 5 1375435 > 4567 25 = S^5 30 = 60x5 25 = 5QOXS 35 = 7000x5 J5 = 30000x5 9628045 = 7 timeitheiiiut- 8152610 = 60 timet do. 6877175 = 500 times do. 5501(740 rr 4000 time do. 187825 = 37565x5 6281611645 = 4567 tknei dow Beside the preceding method of proof, there is another very convenient and easy one by the help of that peculiar property ■ the number 9, mentioned in addition ; which is performed thus : Rule SIMPLE MULTIPLICATION. 2^ iire you are multiplying by« and carrying for the tens, as in addition* 3. Add all the* lihts together, and their sum is the product. Rule i. Cast the Dines out of the two factors, as in additioOf and set down the remainder. 2* Mfiitiply the two remainders together, and if the excess of Htfies la dieir product is equal to the excess of nines in the total |>roduct, &e answer is right. EXAMPLE. 4*15 3==excessof 9^5 in the multiplicand. S78 5=5ditto ih the multiplier. 29505 53720 3700770 6isdItto in the product srexcess of 9*s in 3 Xj*. I>emokstratiomOTtheRule. Let Af and iV he the number of 9's ;n the factors to be muhiplied, and a and h what remains * 6jen M-l-a and N'^B will be the numbers themselves, and their product is MxN-^- Mxb + Nxa+aXb; but the three first of these product are each a precise nanibet Of 9% because one of their factors is so : therefore, these being cast away, there re- iiains only axb ; and if the 9's are also cast out of this, the excess is &t excess of 9's in the total product ; but a and I are thie excesses in the factors themselves, and 4ixB their product ; therefore the rule is true. Q^ E. D. This method is liable to the same inconvenience with that in' addition. Multiplication may ^so, very naturally, be proved by division ; for the prbduct being divided by either of the factors, will evi- dently give the other ; but it would have been contrary to good Method to have given this rule in the text, because the ptjpil if supposed, as yet, to be unacquainted yiith division. f 24 ARITHMETIC. Method of Proof. ' Make tlic former multiplicand the multiplier, and the multiplier the multiplicand, and proceed as before j and if this product is equal to the former, the product is right* EXAMPLES* ... (t) (2) Multiply 23456787454 Multiply 32745654473 by 7 by 234 164197512178 Product- 130982617893 *■ 98236963419 65491308946 Product 7662483146682 3. Multiply 32745675474 by 2. Ans. 65491350948. 4. Multiply 84356745674 by 5. Ans. 421783728370* 5. Multiply 3274656461 by 12- Ans. 39295877532. 6. Multiply 273580961 by 23. Ans. 6292362103.- 7. Multiply 82 1 64973 by 3027. Ans. 248713373271. S. Multiply 8496427 by 874359. Ans. 7428927415293. CONTRACTIONS. h When there are cyphers to the right hand of one or both the numbers to be multipliedt RULE- Proceed as before, neglecting the cyphers, and to the right hand of the product place as many cyphers as are in both the numbers. EXAMPLES* Simple MULTiPLitATiON. 25 EXAMPLES. t. Multiply 1234500 by '7500. "345 75 6i72f 86415 9258750000 the Product. » Multiply 46x200 by 72000. Ans. 3320640000^. 3. Multiply 8x5036000 by 7030b. Ans. 57297030800000. IL When the multiplier is the product g/* two or more num* iirs in the fable. RULE.* Multiply continually by thdsc part8> instead of the whole iiumber at once. EXAMPLES. X. Multiply 123456789 by 25. 123456789 5 61728394s 5 ^3086419725 the Product* - ■ 2. Multiply * The reason of this method is obvious ; for any number mul- tiplied by the component parts of another number must give the same product^ as dlough it were multiplied by that atimber at o&ce : thus in example the second* 7 times the product of 8> mvdtiplied into the given number, makes ^6 times that gireo number, as plainly as 7 times 8 makes 56* D 26 ARITHMETIC. 2. Multiply 3641 1 1 by 56. Ans. 20390216. 3. Multiply 7128368 by 96. . Ans. 684323328. 4. Multiply 123456789 by 1440. Ans. 177777776160. SIMPLE DIVISION. Simple DizLio/i teacheth to find how often one num- ber is contained in another of tl>e sam6 denomination^ and thereby performs the work of many subtractions. The number to be divided is called the dividend. The number you divide by is called the divisor. The number of times the dividend contains the divisor is called the quotient. - - 4 If the dividend contains the divisor any number of ^ ^; times, ai}d some part or parts over, those parts are called; * the remainder. RULE.* I. On the right and left of the dividend, draw a- curved line, and write the divisor on the left hand, and the quotient, as it arises, on the right. 2. Find * According to the rule, we resolve the dividend into parts, and find, by trial, the number of times the divisor is contained in each of those parts ; the only thing then, which remains to be proved, is, that the several figures of th« quotient, taken as one number, according to the order in which they are placed, is the true quotient of the whole dividend by the divisor | wluch may be thus demonstrated : Demon. The complete value of the first part of the dividend, is, by the nature of notation, 10, .100, or 1000, &c. times the value 'SIMPLE DIVISION. 2^ 2. Find how many times the divisor may be had in as many figures of the dividend, as are just necessaryi and ' write the number in the quotient. 3. Multiply the divisor by the quotient figure, and set the product under that part of the dividend used. 4. Subtract value of which it is taken in the operation ; according as there are r, 2, or 3, &c. figures standing before it ; and conseqiyntly the true value of the quotient figure, belonging to that part of the dividend, is also 10, 100, or 1000, &c. times iu simple value. But the true value of the quotient figure, belonging to that part of the dividend, found by the rule, is also 10, ioo» or loco, &c. times its simple value : for there arc as many figures set before it, as the number of remaining figures in the dividend. Therefore this first qaotieot figure, taken in its com. plete value, fi*om the place it stands in, is the true quotient of the divisor in the complete value of the first part of the divi- dend. Fpr the same reason, all the rest of the figures of the quotient, taken according to their places, are each the true quo- tient of the divisor, in the complete value of the several parts of the dividend, belonging to each ; because, as the first figure <)n the right hand of each succeeding pan of the dividend has a less, number of figures, by one standing before it, so ought their quotients to have ; and so they are actually ordered : conse- quently,, taking all the quotient figures in order as they are placed by the rule, they make one number, winch is equal to the sum of the true quotients of all the several parts of the dividend ; and is, therefoie, the true quotient of the whole dividend by the diviscM*. Q^ E. D. To leave no obscurity in tWs demonstration, T shall illustrate it by an example, EXAMPLE. 28 ARITHMETKX T 4. Subtract the last found product from that part of the dividend, under which it standS} and to the right hand of ^e remainder bring down the next figure oi the Divisor 36)85609 Dividend xst part of the dividends 85000 56 X 2000= 72000 2000 the 1st quotknt* ' ■> ■ I St remainder - 1300Q • a4d 6po 2d part of the dividend 13600 36 X 300 = loSoo: 300 Ihc 2d quotlfvt;. 2d rcipainder - 2800 add 00 3d part of the dividend 2800 3(i X 79 = 25^ - rw 7d the .3d quotient. 3d r«;mainder - 289 add 9 • 4th part of the dividend 289 36 X # = 288 - - 8 tkc 4th quotient. Last remainder rr i 2378 ^"^^^.V^JryS!^''* Explanation. It is evident, that the dividend is resolved into these parts, 85000+600+00-1-9 : for the first part of the dividend is considered only as 85, but yet it is truly 85000 ; and therefore its quotient, instead of 2, is 2DOO, and the remainder 1 3000 ; and so of the rest, as may be seen in the operation. When there is no remainder to a division, the quotient is the x^tsolute apd perfect answer to the q^estio|i ; but wher?, there is a remainder, it may be observed, that it goes so much towar^ another time, as it approaches to the divisor j thus, if the remain- der be a fpurth part of the divisor, it will go one fourth of a time more 5 if half the divisor, it will go half of a time more ; >and so on, SIMPLC DtVtSIOK. 2^ the dmdaid ; which number divide a& before ; and . sa o^;,! till the whole is finished* Method on. In order, therefore, to complete the quotient, juA the last remainder at the ^ of it^ abore a small line, and the divisor be« loy^ it. It is sometimes difficult to $nd l;iow often the divisor may be had in the numbers of the several steps of the operation ; the best way will be to find bow often the. ftrst figure of the divisor may be had in the first, or two first, figures of the ^vidend, and the answer made less by one or two is generally the figure wanted : beside, if after subtracting the prodict of the divisor and quo- ti(ent ffom the dividend, the remainder be equal to, or ^xcecd the divisor, the quotient figure mu!||t be increased accordingly. If, when you have brought down a figure to -the remainder, it is stiU less than the divisor, a cypher must be pi|^ ia the quotient^ aod another fi^e brought down^ ^nd then preyed as before. The reason of ^e method of proof is plain : for since the quo^ tient is the n\)mbet of times the dividend contains the divisor, the produa of the quotient and diyisor must evidently be equal to the dividend* Tliere are several other methods made use of to prove division : tjie best aqd most useful are these following. RuL^ I. Subtract the remainder from the dividend, and divide this number by the quotient, and the quotient fojonc^ by this divis* . ioi^ will be equal to the former divisor, when the work is right. The reason of this rule is plain from what has be^a obs^ycd ^bove« Mr. Malcolm, in his Arithmetic, has been drawn into a mistake concerning this method of proof, by making use of particular num- bers, instead of a general demonstration. lie says, the dividend being divided by the integral quotient, the quotient of tills division will be equal to the former divisor, with the same remainder.— fhis is true in some particular cases ; but it will not hold, when the 3*. ABlTHiCETIG. Mftiod of PM(H^. Mtdtiplj the quotient by the divisor^ and this product^ added to the reraainderi will be equal to the dividend^ when the work is rights EXAMPLES^ (I) 5)13545728(2709145-1. (*) 36i)i23456789(33ai37 i<^5 35 35 1395. ioj>5 45 ■ 45. 3006 2920 > 7 5 857 730- 20 137a 1095 28 25 ' *»39 2555 3 %84 3. Divide the l-craaindcr is greater than the quotient, a$ may be easily demon- strated 5 but , one instance will be sufficietit ; thus 1 7^ divided by 6, gives the integral quotient 2, and Remainder 5 ; but 17, di- vided by 2, gives the integral quotient 8, and remainder i. This shews how cautious we oug^to be in deducing general rules from, particular examples. Rule II. Add the remainder^ and all the products of the several quotient figures, by the divisor, together, according to the order, in which they stand in the work, and the sum will be eqiial to the dividend, when the work is right. The SIMPLE DIVISION. 3I 3. Divide 3756789275474 by 2.^ Am. 1878394637737. 4. Divide 12345678900 by 7. Ans. 1763668414-5-. 5. Divide The Feoson of this rule is extremely obvious : &r the oum- bers» that are to be added, are the products of the divisor by eve- ry figure of the quotient separately, and each possesses, by its place» its complete value ; therefore, the sum of (he parts, together with the remainder, must be equal to the whole. Rule III. Subtract the remainder from the dividend, and what remains will be equal to the product of the divisor and quo- tient ; which may be proved by casting out the nines, as was done in multiplication. This rule has been already demonstrated 19 muloplication. To avoid obscurity, I shall give an example, proved according to all the different methods. EXAMPLE. 87)123456789(14^9043 "3456789 87* 87 48 * 364 9933301 Hi9043)"345674i(87Div*SuS[ 348* 1135^344 "35*344 48 • 165 ■■ 9933301 ..87* 123456789 Proof by Mult. 993330I ..786 ..783* . . . . 378 Proof hy tafimg out the nines* .... 348* 4 is the excess of 9's in the quotient. ' ' 6 ditto - - - - in the divisor. 309 6 ditto - -- --in 4x6, which 261* is also the excess of 9's in (123456741 ) — the dividend made less by the remainder. 48^ 123456789 Proof by Addition. For illustration, we need only refer to the example ; except for the proof by addition 5 where it may be remarked, that the asterisms shew the numbers to be added, and the dotted lin<"^ their order. gt 1&RITHMSTIC. 5. Divuie 98765432x0 by 8. Ans. i2j45679oi|i 6. Divide 1357975313 by 9. Ans. 150886145^ 7. Dividc32i7684329765by 17. Ans. 18927*5548809—7. S,. Divide 3211473 by 27. Ans. 1 189434-7. 9. Divide T406373 by 108; Ans. I302i44i. 10. Divide 293839455936 by 81^05. Ans. 3496007 fi^^s^i 1 1. Divide 4637064283 by 57606. AnS; 8049 6-^^^;; tONTRACTIONS. 1; Sn> Svide by any number imth cyph&s Hmne^^m RULE.* Cut oiF tke cyphers ^rom tlie divisoTi and the same number of digits from the right hand of the dividend ; then divide^ making use of the remaining figures, as usual, and the quotient is the answer-; and what remains, writ*^'" ten before the figures cut off, is the true remainder. EXAMPLES. X. Divide 310869CI7 by 7100. 71,00)3108690,17(437844115 the quotient* 284 268 213 "176 .497 599 310 284 2617 . 's. Divide * The reason of this contraction is easy to conceive : for the cutting off the same figures from each, i$ the same as dividing each of siil^LS Diyi^iON. 33 i. bividfe 7380964 by 230oo, • Ans. 3204^ > •9 < 4 z 3 o o o* 3. Divide 29628754963 l^yaS^^oo- ^^^'^^^SSS^jrU' tl; |?^Af« /A^ rfiWfiir is, the product of two or more sinnU mm- bers in Ihe table. RtJLE.* Divide continually by those numbers, instead of the ^hole divisor at once: EXAMPLES. of them by 10, 100, 1000, &c. and it is evident, that ^s often as the whole <^visor is contained in the whole dividend, so often must any part of the divisor be contained in a like part of the dividend. This method is only to avoid a needless repetition of cyphers^ which i^ould happen in the common way, as may be seen by working an example at larg^. • This follows from contraction the second in multiplication, of which it is only the converse; for the third part of the half of any thing is evidently the same as the sixth part of the whole ; and sa of any other number. I have omitted saying any thing, in tlie rule, about the method of finding the true remainder ; for as the learner is fupposed, at present, to be unacquainted with the nature of frac- tions, it would be improper to introduce them in this part of the work, especially as the integral quotient is sufficient to answer mo^ of the purposes of practical division. However, as the quotient ia incomplete without this remainder, and, in some computations, it is necessary it should be known, I shall here shew the manner of find- ing it, without any assistance froto fractions. Rule. Multiply the quotient by the divisor, and subtract the product from the dividend, and the result will be the true re- mainder* ■ - The truth of this is extremely obvious ; for if the product of the divisor and quotient, added to the remainder, be equal to thn dividend, their product taken from the dividend must leave the remainder* The 34 iRITHMETlC. EXAMPLES* I. 'Divide 31046835 by 56^:7X8. . 7)31046835(4435262 ^)4435262(554407 the qtioticat. 28 40 30 4S a8 40 24 35 3^ 3^ 35 J!^ 18 63 14 56 4S ^ 42 — 14 — — ~ 1. Dmrfe The rule which is most commonly made use of is this : Rule. Multiply the last remainder by the preceding divisor, or last but one, and to the product add the preceding reminder ; ibultiply this sum by the next preceding divisor, and to the pro- duct add the next preceding remainder ; and so on, till you have, gone through all the divisors and remsdnders to the first. EXAMPLE. 9}64865 divided by 144. i the last remainder. Mult. 4 the preceding divisor. 4 Add 3 the second reimunder* 450 I 7 ' Mult. 9 the first divisor^^ Add 2 the first remainder* Am- 450^. 6s To "It SIMPLB OXVISION. 35 ;t. DiHdc 7014596 by 72:^;8X9. 8J7014596 9)876824 4 97424 8 the quotient* 3- Divide 5130652 by 132. Ans. 388687^. 4. Divide 83016572 by 240. * Ans, i459^2-^ o" III. Tq perform divism fftorr concisely than hj the general rule. RULE.* Multiply the divisor by the quotient figures as before, ^d subtract each figure of the product as you produce it, i^ways remembering to carry as many to the next figure as wcrq borrowed before^ EXAMPLES. |. Pivide 3104675846 by 833. 833)3 io4675846(3727ioiyxt ^^ qnoticmu 6056 3257 5915 848 713 2. Divide To explain this rule bom the example^ we may observe, that every vuoit of the first quotient may be looked upon as containing 9 of the units in the given dividend ; conseqaendy every unit, that remainS} will contsdn the same ; therefore this remainder must be muluplied by 9, in order to find the units it contains of the ^ven dividend. Again, ev^ry unit in the next quotient will con- tain 4 of th^ 4>receding pnes, or 36 of the first, that is, 9 tiroes 4 ; therefore what remains must be multiplied by 36 ; or, which is the sune thing, by 9 and 4 continually.. Now this is the same as the rule ; fqtr instead of finding the remainders separately, they are re- duced from the bottom upward, step by step, to one another, and the remaining units of the^ same class taken in as they occur. * The reason of this rule^b the same as that of the general rule. 3^ AmnrnifBTiCi 2. Divide 2^137062 by 5317. Ans. 5479 j * ] P -, 3. Divide 62015735 by 7803. Ans. .^^4^^±±, 4. Divide 432756284563574 by 873469. Ans, 495445498|:L|£i^, REDUCTION, TABLES OF COIN, WEIGHT, and MEASURE/ MONEY. 4 farthings make i penny 12 pence i shilling 20 shillings i pound. /or J- d denotes pounds shillings pence. ^- is one farthing, or one quarter of any thing. •J- a half-penny, or a half of any thing. ■| 3 farthings, or 3 quarters of any thing. PENCP TABLE, 20 30 40 60, 70 80 •90 JOG 110 120' is J. I 2 3 4 5 5 10 . 6 7 . 8 9 10 12 24 36 48 60 72 84 96 108 . 120 IS . J". I 2 3 4 S 6 7 8 • 9 10 TROy WEIGHT. . 24 grains make i penny-weight, marked grs. dwt. 20 dwt. I ounce, oz. 1 2 pz. I pound, lb or lb. By this weight are weighed jewels, gold, silver, corn, bread, and liquors. APOTHECARIES' \ REDUCTION. 37 APOTHECARIES' WEIGHT. ao grains make i scruple, marked gr. fc. or 9, 3 fc: <ir 3 I dram . dr. or 5. 8 dr. I ounce oz. or ^. 12 oz. I pound lb or lb. Apothecaries use this weight in compounding their med- icines 5 but they buy and sell their drugs by Avoirdupois weight. Apothecaries' is the san\e as Troy weight, having pn]y some different divisons. AVOIRDUPOIS WEIGHT. 16 drams make i ounce, marked dr. oz. 16 ounces i pound lb. 28 lb, I quarter qr. 4 quarters i hundred weight cwt. 20 cwt. I ton T. By this weight are weighed all things of a coarse or drossy nature : such as better, cheese, flesh, grocery wares, gn4 all njetals, except gold and silver.* DRY lb. • A firkin of butter . is . $6 A firkin of soap 64 A barrel of pot-ashes . . . 200 A barrel of anchovies .... 30 A barrel of candles .... 120 . A barrel of soap 256 A barrel of butter 224 A fother of lead is 19^ cwt. A stone of iron 14 A stone of butcher's meat . . 8 A gallon of train oil . . . . 7^ A faggot of steel 120 A stone of glass 5 A seam of glass is 24 stone, or 126 lb. oz. dr. A peck loaf of bread weighs 17 61 A half peck .... 8 11 A quartern .... 4 58 56 lbs old hay! J make a truss. 60 lbs new hay ^ 36 tiusses a load. 4 pecks coal make i bushel. 9 bushels ... I vat or strike. 36 bushels .... 1 chaldron. 21 chaldrons . . . i score. •7 lbs wool make . . i clove. 2 eloves I stone. 2 r>tones 38 ' 4iRITHMETIC. PRY MEASUR?, Marked 2 pints make i quart pts.qts. 2 quarts i pottle pot. 2 pottles 2 gallons 4 pecks I bushel 2 bushels % I strike' I I gallon X peck gd. pc. bu^ str. Marked 8 bushels i quarter qr. 5 quarters j weycrload wey 4 bushels i coomb co. 5 pecks z bushel water meaf» 10 coombs I wcy 2 weys I last L, Note. — ^The diameter of a Winchester bushel is i8-|- inches, and its depth 8 inches. By this measure, fait, lead, ore, oysters, corn, and othe» dty goods are raeafurcd. ALE AND BEER MEASURE. Macked 2 pints make i quart pts. qts. 4 quarts I gallon gal. 8 gallons i firkin of Ale fir. p gallons I firkin of Beer fir. 2 firkins 2 kilderkins Marked I kilderkin kil. I barrel bar. 3 kilderkins i hogshead hhd**' J barrels i butt butt;. Note.— The ale gallon contsuns 282 cubic inches. In London the ale firkin contains 8 gallons, and the beer fir<» kin 9 5 other measures being in the same proportion. WINE 2 stones • . ' I tod. 6t tods 1 wey. 2 weys I sack. 1 2 sacks I last; , lb. A barrel of pork 15 ... . 220 A barrel of beef ..... 220 A quintal of fish 112 20 things make . • . • i score 12 I dozen. 12 dozen .1 gross. 144 dozen . . . i greater gross. i^tfr/ifcrr— 5760 grains =: i lb. Troy ; 7000 grains = I lb. A- Toirdupois ; therefore the weight of the pound Troy is to that of the pound Avoirdupois, as 5760 to 700a, oi[ as 144 to 175. kfeDUCTION. Wine m e a a u r Marked ft pixite make l quart pts^ qts. 4 quarts i gallon 4^ gallons i tiefce tier. 6^ gallons I hogshead hhd. $4 gallons i puncheon pun 39 E. Markfi I pipe or p. or h. X pipes I tun T. 1 8 gallons i rundht rund. 314- gallons I barrel bar. 2 hogsheads butt By this measure, brandied, spirits, perry, cider, mead, Vinegsir, and oil are measured. NotE.— 231 solid inehes make ^ gallon, and 10 gallons tnake an anchor. CLOTri MEidSURfi. ' I^ked 2^ inches make i juail nls. 3 qrs. 4 nails I quarter qrs. 5 qrs. 4 quarters t yard yds. 6 qrs. LONG MEASURE I ell Flemish EllFl. X ell English EllEng. 1^11 French Ell Ft. Marked 3 b^ley corns make i X inch bar.c in. 12 inches i foot ft. 3 feet X prd * yd. 6 feet 2 fathom ^||^. 5f yards 1 pole pol. 40 poles I furlong fur. 8 furlongs i m3e mis. 3 miles I league 1. Marked 60 geographical ifyiks,or 6^ statute mile^ t de- gree deg. or * 360 degrees the circum- ference of the earth. Note — 4 inches make i hand. 5 feet I geometrical pace. 6 points 1 line ' 12 lines X inch- T I M E. 60 seconds make 1 min ute 60 minutes 24 hours 7 days Marked s. or " m. or ' X hour h. or ® X day d. X week w. 4 weeks x month 13 months, i day, and 6 hours, or 365 days and 6 hours, x Julian year Marto^d m. T. Note. 40 ARITHMETIC. Note i. The second maybe supposed to be divided into 60 thirdS) and these again into 60 fourths, &c. Note 2. April, June, September, and November, have each 30 days ; each of the other months has 31, except February^ "which has 28 in common years, and 29 in leap years. CIRCULAR MOTION. 60 seconds make i minute, marked '' ' 60 minutes i degree ® 30 degrees i sign s; 12 signs, cr 360° i circle^^ keduciton is the method of bringing numbers from onef name or denomination to another, so as still to retain, the same value, . IIULE.* I. When the reduction is from a greater name to a less 4 Multiply the highest name or denomination by as many as make one of the next less, adding to the product the parts of the second name ; then multiply this siim by as many as make one of the next less name, adding to the product the parts of the third name ; and so on, through all the denominations to the last. ^ II. When * The reason ,of tliis rule is exceedingly obvious ; for pounds are brought into shillings by multiplying them by 20 ^ shillings in- to pence by multiplying them by 1 2 ; and pence into farthings by multiplying them by 4 ; and the contrary by division : and this will be true in the reduction of numbers consisting of any denom- ination whatever. ^ ktouctioit 41 It. When the reduction is from a less iUUae io agreaten DiVide the given nnmber hj as many as make one of the next superior denomination ) and this quotient again by as taany as make one of the next following j and so on^ through all the denominatiotis to t^e highest ^ and this last ^quotient, together with the several remainders^ will be tbi iiUDAwer required. Themediod of proof is by reversing the question^ SXAUPLSS. i. la 1465]. 14^. jd. how many farthings ^? 20 ^4)1407092 29314 12)351773 12 3SI773 2,0)2931,4 S i * ■ 4 Proof 1465I. 14s. $dm J 407092 tlie ansiHrer^ 2. lii 12L how many farthinjgs^ Ans. 11520* 3* In -6x69 pence how many pounds t Ans. 25I. 14s* id. 4. In 35 guineas how many farthings sterling ? Ans. 35J180. J. In 420 quarter guineas how many moidores ? Ans. 81 and 1 8s. 6. In 23 1 1. x6s. how many ducats at 4s. 9d. each ? Ans. 976. 7. In 274 marks^ each 138. 4d. and 87 nobles, each 6s. Id. how many pounds i Ans. 2ill. 13s. 4d. 6. In i77(rquaTter guineas how many six-pences sterling? Ans. 18648. ^ 9. Reduce 1776 six and thirties to half-crowns sterling. Ans. 2S574T- xo. In ^9807 njioidoKs how many pieces of coin, each 4S.6d. ? ■ A|is* 304842. f XX. In 44 AKITQMETIC* TROY WE IG HT. 9>* oz. dwt. gr. Jb. oz. dwt. gr. lb. oz. dwt. gr. 17 3 'S i» 14 10 13 20 27 10 17 18 13 2 13 13 13 IP 18 21 17 10 »3 »3 15 3 «4 14 14 xo to 10 13 " 13 ' 13 <0 10 I 2 3 10 I a i« ' I U 1444 4 4 3 3 X3 »4 I 19 2 .1 . APOTHECARIES' W E I G H T. ib. oz. dr. K. ff. lb. OS. dr. 8c gr. Rv (»• dr. 1 •c. gr- 3 5 7 « 17 4 5 ^ X »3 5 4 3 I 19 a 7 4 2 18 2 7 5 2 17 I I 2 7 43a z 18 I 7 5 I 10 3 a I 1 17 I 7 5 *■ »o 3 4 2 1 4 4 a I I 4 » 7 3 * »7 2212 3 * IP 3 6 I I IP 3x11 I 7 » ^ AVOIRDUPOIS WEIGHT. cwt. qr. ll>. oz. dr. T. cwt. qr. lb. oz. dr. T. cwt. qr. . lb. oakdiw 15 2 15 IS IS 2 17 3 13 8 7 3 13 a 10 7 7 13 2 17 13 M 2 13 3 14 8 8 2 14 I 17 6 6 12 2 13 14 14 1 16 10 5 4 17 14 6 10 I 17 IS 2 13 17 2 13 12 7 7 12 I 10 10 1 14 I 12 2 3 13 10 4 4 10 I 12 I 7 4 16 I 7 7 s S a 12 8 8 LONG eOMPOUMV AOVITI0K. 4jr LONG. MEASURE. ]Mbikr.pnl.yd.ft in. Mis. far.pol. yd.ft in. ' MIsAr.pol. yd.ft. in. 37 3 14 2 I s 28 2 13 I I 4 28 3 7 2 7 28 4 17 3 2 10 39 I 17 2 2 10 30 I 7 17 4 4 3 I 2 28 I 14 2 2, 27 6 30 2 2 - to 5631 7 48 1 17 227 76 20 21 29222 3 3T I ^9 3 52 2 10 yx 42 22021 7 10 22 CLOTH M E A S U R E- Td. qr. nL in. I^ll En. qr. nl. in. Ell FL qr. nl. in. 120 3 I I 207 2 2 I 200 2 I I 6» 2 1 ^S 2 2 57 I I 28 2 2 7« I I 28 I I I 3» 2 2 21 3 3 2 21 2 « 2 3 20 2 2 38 3 1 18 3 2 2 3 2 2 2 WINE MEASURE. X' bh<t gaL qt. pt. T. hhd. gal. qt. pt. T. hhd. gaL qt. pC. 17 2 10 2 I 27 I 3 I I 37 I ^ I I 10 2 27 2 I 24 13 I 27 27 3 1 8 3 24 2 21 3 37 20 2 24 5 2 27 2 10 2 35 I I ^O I 29 2 I 211711 82 25 II 3 39 21 3 29 2 I 2 2 35 2 2 37 a I ALS 42 ' A&tTHMftTlC. IX. In 2x3210 grains how many lb. I Am. 37lb. jugxsr 12< In 591b. I3dwt8. 5gr. how many grains ? Axis. 340i57gis* 1^3. In 80x2x31 grains how many lb.? Ans. X39olb. xi07» xSdwts. x^gu. 14. In 35toni X7cwt. iqir. 231b. 7oz.,X3dr. how many drama ? Ans. 2Q57Xoo5dr. * X 5. In 3 7cwt. 2q[r. x 71b. how many pounds Troy, a pound Avoirdupois being equal to 140Z. iidwt. X5~gr8.Troy ? Ans. 5 x241b. 50Z. xodwt xi^-grs* t6. How many barleycorns will reach round the worlds supposing it> according to the best calculations, to be 8340 leagues ? Ans. 475 5 80 1 6oo. 17. In 17 pieces of doth, each 27 Flemish ells, how many yards ? Ans* 344yds. xqr. . 1 8. How many minutes were there from the birth of Christ to the year 1776, allowing the year to consist of 365d. 5h. 48' 58" ? Ans. 934085364^^ 8'^ COMPOUND ADDITION. -h Compound Addition teacheth to collect several numbers of different denonmmtions into one total. RULE.* X. Place the numbers so th^ those of the same denom- ination may stand directly under c^h other, and draw: a line bdow them. 2. Ad4 * The reason of this rule is evident from what has been said in simple additton : for, in addition of moaey^ as i in the pence is «q«al toMPOTOD ASDinOX. 43 1. Add up the figures in the lowest denomination, and find how many ones of the next higher denomination are contained in their sum. 3. Write down the remundcfi. and carry the ooes to the next denomination $ with which proceed as before ( and so on^ through aU the denominations to the highesti ^ose sum must be all written down ; and this sum, together with the seyeral remainders, is the total sum required* The method of proof U the same as in simple addition* eXAKPLES. p M N E V. :£• a. i. 75 s« d. t s> i. »7 »3 4- »7 13 5k 5* »75 107 10 13 10 13 10 2' "1 10 »7 3 5« 17 8i 89 18 10 8 8 7 30 10 lot 75 12 2i 3 3 4 17 '5^ 4i 3 3 3i 8 8 10 10 ir r 452 19 i 54, I 8 4 261 5 8t ' n 36 176 8 24 277 8 Ai 54 I 4 261 5 8t 45* 19 n TROY equal to 4 in the farthings ; i in the shilliogSi to i a in the pence ; and I in the pounds, to 20 in the shillings ; therefore, carrying as directed, is nothing more than proriding a method of digesting the money, arising from each column, properly in the scale of de- wlmiinations ; and this reasoning will hold good in the addition of compound numbers of any denomination whatever. fj| ABITHMBTIC^ TROY WEIGH T^ lb. oa. dwt. P- lb. oz. dwt. gr. lb. oz. dwt. «f- 3 '5 11 14 10 13 20 27 10 »7 18 a 13 13 13 10 18 21 17 10 »3 »S 3 «4 14 14 10 10 10 »3 II «3 I. |o 10 I a 3 10 I « I 17 1444 4 4 3 3 13 14 X 19 APOTHECARIES' WEIGHT. )b. oz. dr. tc. gr. lb. os. dr. sc gr. Ibu oe. dr. ic. gr. 357^ I? 45^1 »3 54311^ 2742 18 275a 17 4322 la 17511a 1612 7 321117 175 2- xo 3421 4 4211 4 ^732 17 2212 32 IQ a 6 I I ip 3 I I I I T z % t ■ ■' J «. ■ 'J W.J lU AVOlRDyPOIS WEIGHT. cwt.qr. H). oz. dr. T. cwt. qr. lb. oz.dr. T. cwt. qr. lb. o&dfsn 15 2 15 IS IS 2 17 3 13 8 7 3 13 2 «o 7 7 13 2 17 13 14 2 13 3 14 8 8 2 14 I 17 6 6 12 2 13 14 14 I 16 10 5 4 17 14 6 10 I 17 IS 2 13 I 7 2 13 12 7 7 12 1 10 10 I 14 t I 2 2 3 13 10 4 4 10 I 12 I 7 4 16 I 7 7 5 5 2 12 8 8 LONG ^ eoMPOofMD Avamov. 4^ ■LONG. MEASURE. yS^Air.pnl.yd.ft in. Ml». ^ar.pol. 7d.ft in. ' MIsArr.pd. yd.ft. in. 37 3 14 2 I S 28 2 13 I I 4 28 3 7 2 7 28 4 17 3 2 10 39 I 17 2 2 10 30 I 7 17 4 4 3 I 2 28 I 14 2 2 2763022 1056317 48 1 17 227 76 20 21 29 2 2 :i 3 37 I 29 3 52 2 10 ^ 42 22021 7 10 22 CLOTH M E A SURE. Td. qr. nL in. I^IlEn. qr. nl. in. Ell FL qr. nl. in. 120 3 I I 207 2 2 I 200 2 I I a* 2 I 58 2 2 57 I I 28 2 2 78 I I 28 I I I 39 2 2 21 3 3 2 21 a 5H^ 2 3 20 2 2 38 31 i« 3 2 2 32 2 2 WINE MEASURE. 7*. bh<t £aL qt. pt. T. hhd. gil. qt. pt. T. hhd. gal. qt. pC. 17 2 10 2 I 27 I 3 I I 37 I 2 I I 10 2 27 2 I 24 13 I 27 27 3 1 S 3 M 2 21 3 37 20 2 2+ 5 2 *7 2 10 2 35 I I 3JO I 29 2 I i I 17 I I 8 2 25 I I 3 39 2 I .3 29 2 I 2 2 35 2 2 37 2 I ' AI.S 46 ARITBlIETie. ALE AND BEER MEASURE. bhi. gaL qt. pt. hhd. fd.qt.pt. hhd. 8»L y, . pt. 31 2 2 I *7 3 « » 30 20 3 I 21 20 3 »5 lO s s8 29 3 21 21 2 21 «3 20 SO lO lO 2 lO >7 x8 18 ■I 1- i 3 3 8 7 » 17 «7 2 2 — ,JL 2 '2 I 6 6 I ■ I DRY MEASURE^ T 1 M E. "■ . . . — In, qr. biL pe. g«l. Y. m. w. d. h. m. t* 5 J * 3 ^ 27 9 » 6 23 »5 *S S » 3 2 I 20 7 2 5 20 36 30 2 2 3 2 X »« 7 3 4 5 6 7 ? 2 2 14. J 31 22 *3 2 I 7 10 2 4 5 S 5 6 « 8 2 4 3 i! \ COMPOUND SUBTRACTION. Compound Subtraction teachctji to find the difierence of any two numbers of different denominations. kULE.* I. PlaAe the less number under the greater, so that those parts,, which are of the same denomination, may stand di- rectly under each other^ and draw a line belo^ them. 2. Beg^ * The reason of this rule will readily appear from what was said In simple subtraction ; for the borrowing depends upon the TCry «»me principle, and is only different, as the numbers to be subtracted are of different denominations. COIIPOVHD lUBT&ACTION. 47 d« Begin at die riglit liand» and lake the number in each denomination of the lower line from the number standing above itj and set down their remainders below Aein« 3. But if the number below be greater than that above it, increase the upper number hj as many as make one of tlieiiext higher denomination^ and from uiis sum take the number in the lower line^ and set down the remainder as before. 4* Carry the unit borrowed to the next number in the lower line, and subtract as before ; and so on, till the whole is finishM 1 aind all the several remainders taken together, as one number^ will be the whole difference re- quired. The method of proof is die same as in simple suk traction. exiUPLES. M N E Y. From 275 '^aU 176 s. 16 d. £. s. d. 454 14 2i 276 17 si 274 85 8. »4 15 d. 7i Rem. '98 16 10 177 16 9i 188 18 6i ftwof 275 »3 4 454 »4 H 274 «4 2t - TROY WEIGHT. R>. oz. Take 3 7 , dwt. >4 II 2Q lb, oz. dwt. gr. 27 2 10 20 20 3 '5 21 lb. oz. ap 3 20 7 dwt 14 >5 gr. 5 7 Rem. Proof AFOTHECARIES* .t «8 ' tek'raik&'nc. APOTHECARIES' WEiGii*r. lb. oz. dr. %. gr. )b. 6z. dr. sc. gr. lb. oz. dr. sc. gh ^rom 1 1 4 7 14 s 3 6 z 10 5 x 3 1 19 Take ST^-^S i87»i2 225. i I i I "i • 'i I • ' ■ I I II I ■ m Rem. ^■y Proof A-^V O I Jt D U POIS WEIGHT. twu qK lb. ozk dr. cwt. qnlb^ oz.dr; cwt. qr. lb. oz. dr^ From 5 17 5 9 22 2 13 4 8 21 I 7 6 13 Take 3 3 21 i 7 ao i 17 6 6 13 8 8 14 tl( .ea. ftoof LONG MEASliRfi. Mlt. for. poL yd. ft. in. lAls, fur. poL yd. ftjn. Mitf. fur. poL yd. ftiiL From 14 3 x; I 2 z 70 7 13 z x 2 70 3 zo ^ Takexo 7 30 2 zo 20 Z4 » 2 7 X7 3 zz z z 7 'I I ■> ■ ■■ I ii I • ■ • 1 1 II ilem. iVoof CLOTH MEASURE. Yd. qr. nl* Ell En. qr. nL Ell Fl. qr. nl. in. From 27 3 3 127 2 270 I I Take xo 2 2 78 3 3 140 222 Rem. Proof WINE ^ COMPOUNB SUBTEACTIOM. 49 WINE MEASURE. T* hhd. gaL qt« pc hhd. gal. qt. pU tihd. gaL qt» Vrotn 2 3 20 3 I 2 II 2 13 t "^ "~ " 3 I 10 27 X Take I 2 *7 Rem. ^toot ALE AKD BEERMEASU&E. hhd. fir. gsi. qt. pt. hhd. fir* gaL pt hhd. fir. gaL pt, Ffoitt 27 2221 29234 27 322 Tike xo 3 4 3 20 2 4 5 10 } Renu MMB^VmMMMtaMlMMikMMk ^MWi^Maiw4»ilM«MMtalM MMM««iaHMMana*aii^t Phxrf DRYMEASURE. L. qr. hcupe. gaLpot L. qr. bu. pe.gaL I^ qr. bu. pcga!. Ttomg 47111 13 3521 17 12 Take 2 53 7 237 10 2211 ■ —————— ■ ■ *' ' " ■ ■' ■ j" i Rem* Proof TIME. in. w. d. h. ' m, w. d. h. ' m. w. d. h. ' Frdm 17 2 5 17 26 37 i 13 x 71 5 Take 10 18 x8 15 2 15 14 17 5 5 7 Rem* Fooof COMPOUND I 50 . ARITHMETIC. COMPOUND MULTIPLICATION. Compound Multiplication tcacheth to find the amount of any given number of different denominations by repeating, it any proposed number of times. RULE.* T. Place tKc multiplier under the lowest dfenominatioa' of the multiplicand. - a. Multiply the number of the lowest denomination, by the multiplier, and find how many ones of the next higher denomination are contained in the product. 3. Write down the excess, and carry the ones to the product of the next higher denomination, with which pro-" ceed as before \ and so on, through all the denominations to the highest, whose product, together with the several excesses, taken as one nttober, will be the whole amount^ required. The method of proof is the same as in simple multi- plication. EXAMPLES * The produa of a nnmber consistiBg of several parts, ot d^ nominations, by any simple number whatever, will evidently be' expressed by taking the product of that simple number and each' part by itself, as so many distinct questions : thus, 25I. 12s. 6d. multiplied by 9 will be 225L io8s. 54d. = (by taking the shillings from the pence, and the pounds from the shilliAgs, and placing, them in the shillings and pounds respectively) 250L I28. 6d. which is the same as the rule ; and this will be true^ when the moltiplicand is any compound number whatever. COMPOUN© MULTIPtlCATION. ^l EXAMPLES OF MONET. I. plb. of tobacco^ at 2s. S^d. per lb. il. 48. 4id. die answer. ru 31b. of green tea, at ps. 6d. per lb. Ans. il. 8s. 6d. 3. 51b. of loaf sugar, at is. 3d. per lb. Ans. 61. 3s. 4. pcwt. of cheese, at il. i is. 5d. per cwt. Ans. 14I. 2s. pd. 5* 12 gallons of brandy, at 9s. 6d. per gallon. Ans. 5I. 14s. CASE I. J^ the mtiltiplter exceed 12, multiply successively by ks component parts, instead of the whole number at once, as t in simple multiplication. EXAMPLES. 9. i6cwt. of cheese, at il. 18s. 8d. per cwt^ jl. 18s. 8d. 4 7 14 8 4 ^"30 18 8 the answer. 1. 28 yards of broad cloth, at 19s. 4d. per yard. Ans. 27I. IS. 4d. 3. p6 quarters of rye, at il. 3s. 4d. per quarter. Ans. II 2I. 4« 120 dozen of candles^ at 5s. pd. per doz. Ans. 34!. I OS. 5. 132 yards of Irish cloth, at 2S. 4d, per yard. Ans. 1 51* 8o. 6. 144 reams of paper, at 135^ 4d« per ream. • Ans. 961* 5^ ARrrHlfETie, 7. I2I0 yards of shaUoonj at 28. ad, per yard. Ans^ 1311. ;s. 2dp CASE II. J^ thi ntuhipliir eanmt be produced by the multiplication of small numbers, find the nearest to it, either greater or less;^ which can be so produced ; then multiply by the compo- sept' parts as before $ ^nd for the odd parts, add or sHb« ^ct according as is required. ^XAMPLpSt f. xy elk of hoUand, at ys. S^. per ell, 78. 8id, 4 I JO 10 4 <5 3 7 U £6 II o4 the tnsYTtr.^ a. 23 ells of dowlas^ at is. 6ji. per ell. Ans. il, 15s, s-Jd, 2^ 4d' bushels of F^^eat, at 4$. ^^d. per bushel. Ans. lol. IIS. 9|<lt 4. 59 yards of tabby, at ys. lod. per yard. Aqs. 23I. 28. 2d. 5. 94 pair of silk stockings, at 12.S. 2d. per pair. Ans. 57I. 3s. 8d. 6. ii7evt. of Malaga raisins, at il. 2S. 3d. per cwt. ' ' Ans. 1301. 3s. 3d. EXAMPLES PF WEIGHTS AND MEASURES. lb.oz.dwt.gr. ]bwpz.dr. 9C.gr. c:(!rt. qr. lb. pE. mis. fur. pit. ^d^ 2; I 7 13 a 4 2 I 27 . I 13 12 24 3 ao 2 4 7 12 6 ' ■ ■ ■'■ ^ I . ■ ■ ■ ■ ■ . . ■' J.. ' ■ ■ " COMPOUND DIVISION. 53 ydfc qr. dIs. T. bhd. gal pt. W. qr. bu. pe M. w. d. li. \tL J27 2 2 29 I 20 3 27 I 7 2 175 3 6 20 50 857 u COMPOUND DIVISXONV Compound Division teachctli to find how often one given PUqiber is contained in another of (lifFcrcnt dcnoiuinations, RULE.* I. Place the numbers as in simple division. if Begin at the left hand, and divide each denomination fcy the divisor, setting tl\Q quotients .under their re^pccdvf^ dividends, 5» But if there be a remainder, after dividing any of the 4enominations except the least, find how n^any of the next }ower denomination it is equafto, ind add it to the nun-* |)er, if any, which was in this denomination before ; then iJivide the sum gs usual, and so on, t:ill the wliok is finished. The method of prpof is tlie same as in simpTe division, EXAMPLES * To divide a camber coDsiji^D^ of several dcDomiDations, by any simple number \vhatcvcr, is evidently tbc same as dividing all the parts or members, of which that number is composed, by the same simple number. Apd tbis will be trye, when any of the parts are. not an exact multiple of the divisor : for by conceiving the number, by which it exceeds that multiple, to have its proper ^value by being placed in the npxt loM^rer denomination, the divi- dend will still be divided into parts, and the true quotient found as l^fore : thus 25I. 12s. ,3d. divided by 9, will be the same as 18I. 144s. 99d. divided by 9, which is equal to 2I. i6s. iid- as bv ' tj^e fule ; gnd the method of carrying from one denomination to another 19 exactly the same. 54 jlltlTHMBTlC. • EXAMPLES OF MONET. % Divide 225!. 28. 4d. by 2* 2)2251. 2s. 4d. 112L IIS. 2d. the quotient. %. Divide 75 il. 14s, ^^. by 3. Ans. 2jol. us, 6^^ 3. Divide 82 il. I7S» p^d. by 4. Ans. 205I. 9s. 5-|d. 4. Divide 28I. 2S. i^. by 6. Ans. 4I. 13s, 8-^d. 5. Divide 135I. 10^. 7d. by 9. Ans. 15I. is. 2d. 6. Divide 227I. los. 5d, by ii. Ans. 20I. 13s. 8d. 7. Divide 13321.^118^ 8-^. by 12. Ans. ml. ii~d» CASE J. If the divisor excfd 1 2, divide continually by Its com* Bonent parts^ as in simple division^ EXAMPLES* It What Is cheese per cwt^ if i<Jcwt» cost 30I. i8s. 8d. ? 4)301. 1 8s. 8d. '■■ ' " ■ ' « ' ■ * 4)7 14 8 * ^ £,\ 18 8 the answer. 2. If 2ocwt. of tobacco comes to i2olr ips. what if that per cwt. ? Ans. 61. 6d. 3. Divide 57I. 3s. 7d. by 35, Ans. il. 12s. 8d. 4. Divide 85I. 6s. by 72. Ans. il. 3s. 8^. ^ 5. Divide 3 il. 2S. lo^d^by 99. Ans. 6s. 3|.d. d At 1 81. 1 8s. per cwt, how much per lb. ? Ans. 3s. 4-j^. CASE n. If the divisor canmi he produced hj the multiplication of small number Sy divide it after the manner of long division. EZAMFtES* eOMPOUND DIVISIOM. 5^ EXAMPLES. t^ Divide 74I. 13s. 6d. by 17. 17)74 ^3 6(4 7 10 68 6 ao 1 133 119 14 12 174 »7 4 2. Divide 23I. 15s. 7-^. by 37. 3. Divide 3151. 33. lOjd. by 365.- Ans/ 128. io|d.- Ans. 17s. 3-Jd. EXAMPLES OF WEIGBTS ikJAD MEASURES. . I. Divide 23ID. 70Z. 6dwt. lagr. by 7. Ans. 31b. 402. ^dwt. I2gr» a. Divide 131b. loz. 2dr. logr. by 12. Ans. lib. loz. 2SC. logr. 3* Divide io6icwt. aqrs. by 28. Ans.* 37cwt. 3qrs. i81b. 4. Divide 375mls. 2 fur. 7pls. 2yds. ift. 2in. by 39. Ans. pmls. 4fur. 39pls. 2ft. 8in« 5* Divide 571yds. 2qrs. ml. by 47. , Ans. 12yds. 2nls. 6» Divide 120L. 2qrs. ibu. 2pe. by 74. Ans. iL. 6qrs. ibu. 3pe. 7U Divide laomo. 2w. 3d. 5h* 20' by iii. Ans. imo. 2d. loh. 11!. DUODECIMALS- %S A&XTHMETlC. DUODECIMALS. DxTODECIMALS are so ciailed because thejrMccrease hf twelves, from the plate of feet toward the right hand. Inches are sometimes called primes, and are marked thus ^^ tlie next division, after inches, is called parts, or seconds, and is marked thus '^ ; the next is thirds, and marked thus ^'^ y and so on. Duodecimals are commonly used by workmen and ar« tificers in casting up the contents of their work. "^ Multiplication^ of Duodecimals ; or, Cross Multiplication. RULE. t. Under the multiplicand write the sam# names or de« nominations of the multiplier ; that is, feet under f^et, inches under inches, parts under parts, &c. 2. Multiply each term in the multiplicand, beginning at the lowest, by the feet in the multiplier, and write each result under its respective term, observing to carry an unit for every 12, from each lower denomination to its next superior. 3. In the sam^ manner multiply every iprm in the mul- tiplicand by the inches in the multiplier, and set the re- sult of each term one place removed to the right of tlicwe in the multiplicand. 4. Proceed in like manner with the seconds and all the rest of the denominatiohs, if there be any more 5 and the*^ 6U{n of all the lines will be the product required. Or i>0Oi>EcnfAMw 57 Or the denominations of the particular products will ic as follows : Feet by feet, give feet. Feet by primes, give primes. Feet by seconds, give seconds. &c. Primes by primes, give seconds* Primes by seconds, give thirds. Primes by thirds, give fourths^ &c, Seconds by seconds, give fourthsf Seconds by thirds, give fifths. Seconds by fourths, give sixths^ &c. Thirds by third!^ -give sixths. Thirds by fourths, give sevenths^ 'JTiirds by fifths give eighths. In general thus : "WTien ffect arc concerned, the product is of the same 4etionunation with the term multiplying the ieet. When feet are not copcemed, the name of the product will be expressed by the sum of the indices of the twoi factors. .- .'. EXAMPLES. I, Multiply lof. 4' S" by 7f. ^ 6'\ 7^6 72 6' II d 10 II 4, S 2 2 6 ' 79 II 6 6 Answer. 1. Multiply 4f. Y by 6i. 4' o'\ Ans. spf. o'^4'^^ f 3. Mujtif)ly H S9 , «mmETic. 3. Multiply I4f. 9' by 4f. 6'. Ans, 66L 4' 6'^ 4. Multiply 4f- 7' 8" by pf. 6^ Ans. 44f. o' io"« 5. Multiply 7f, 8' 6" by lof. 4' S-^. Ans. 79f. ii'o"6''^6ir, 6. Multiply 39f. 10' 7" by i8f. 8f 4'^ ■* ^ Ans. 745f.6'io''2'^'4iv^ 7. Multiply 44f. 2' 9'^ 2''f 4iv. by 2f. 10^ 3". Ans. I26f. 2^ lo" 8''' i(Mv. iiv* 8. Multiply 24f. 10' Z" i'' siv. by <){. 4' 6''. Ans. 233f. 4' 5" 9"^ 6iv. 4V. 6vi, 9. Required the content of a floor 48fa 6f long, and 24f. 3' broad. • • * ' •• Ans. ii76f. i' 6''. 10. What is the content of a marble slab, whose length js 5f. 7', and breadth if. lo'^ ? ' Ans. I of. 2' \(J'. 11. Required the content of a ceiling, which is 43f. 3' long, and 25f. 6' broad.' - - ^ Ans. iio2f. \cf 6^'. 12. T^e length of a room being 26f. its breadth i4f, ^\ and height lof. 4', how many yards of painting are in i;:, deducting a fire place pf 4f . by 4f. 4', and two win* dows, each 6f. by 6f. 2'?,* io^y ^» * •' "» ''^ Ans. 73-~7 yards. 13. Required the solid content of a wall 53f. 6' longi X2f. 3' high, i;^d 2f. thick. ' ~ . ^ ' * • . ' ^^jj^^ i3iof, 9', VULGAR Vulgar fractions: Fractions, or broken numbers, are expressions for ^7 assignable parts of an unit ; and are represented by two numbers, placed one above the other, with a line drawn bctwpen them. . The figure above the line is called the numerator'^ and that below the line, the denominator. . The denominator, shews how many, parts, the integer i* divided into^ and the numerator shews how many of ihose t>art8 are meant bjr the fraction. Fractions are either proper, improper, single, compound^ or mixed. ...... I. A j)roper fraction is when the numerator is less than the denominator : as -I, ^, -f^, &c. 2* An improper fraction is when the numerator exceeds the denominator : as y, ~^, &c. . 3. A single fraction is a simple expression for any num- ber of parts of the integer. 4. A compound fraction is the fraction of a fraction : as I- of f, i of f , &c. 5. A mixed number is composed of a whole number and a fraction : as 8j, 17-^3-, &c. > Note.— Any whole number may be expressed like a frac- tion by writing i under it : as \. . 6. The common measure of two or more numbers is that number, which will divide each of them, withouc a remainder. Thus 3 is the common measure of 12 and 15 % and the greatest number, that will do this, is called the greatest common measure. , 7. A number, which can be measured by two or more numbers, is called their common multiple ; and if it be the ievist 6b AKITHMETIC. hast number, \7hich can be so measured, it is caQed theil' ' leait common vjuliip/e ; thus 30, 45, 60 and 75, are multi-« pies of 3 and 5 ; but their leadt common multiple is 15** ?ROBJL£M I. To jind the greatest common tneasurt of two or more numberu RULE.f I, If there be two numbers only, diTide the greater by the less, aiTd this divisor by the remainder, and so on > always dividing the l^t divisor by the last remainder, till nothing. ■^ A prime numher is that, which can only be measured by aa emit. That number, which is. produced by multiplying several numbers together, is called a composite numher, A perfect numher is equal to the sum of all its aliquot parts. The following perfect numbers are taken from the Pctersbur^h jictSx and are all that are known at preseou 38 496 8128 8589869056 13743S691328 2305843008139952128^ 2417851639228158837784576 990352031428297 1830448816128 There are several other numbers, which have receired different denominations, but they are principally of use in Algebra, and the ' higher parts of mathematics. f This and the fallowing problem will be, found very useful in the doctrine of fractions, and several other parts of Arithmetic. ^ The truth of the rule may be sh^wn from the first example. — " For smce 54 measures io8, it also measures 108+54, or 162. Again, VUICAK FRACTIONS. 6t iiotking remains, then will- the last divisor be the greatest common measure required. 2. When there are mojc than two numberSf find the greatest common measure of two of them as before ; and. of that common measure and one of the other numbers $ and so on, through all the numbers to the last ; then will the greatest common measure, last found, be the answer. 3. If I happen to be the common measure, the given numbers are prime to each other, and found to be incom- Ihensurable. Examples. t. Required the greatest common measure of 918, 199S fud 522. 918)1998(2 So 54 is the greatest common measure 1836 of 1998 and 918. Hence 54)522(9 162)918(5 486 810 36)54(1 108)162(1 36 108 18)36(2 54)108(2 36 108 * Therefore 18 is the answer required. 2. What Again, since 54 measures 108, and 162, it also roeasuresi 5X162 + 1089 or 918. In the same manner It will be found to measure 2x918 + 1 62, or 1 998, and so on. llserefbre 54 measures both 918 and 1998. It is also the greatest common measare ; for suppose there - be a greater, then since the greater measures 918 and 1998, it also measures the remainder 162 ; and since it measures 162 and 918, it also measures the remainder 108 ; in the same manner it will be found to measure the remainder 54 ; that is, the greater meas- ures 6% JilltTrtMfiTlC. a. What id Hit greatest common measure oi 6il aha 54^ ' Ans. 36-. 3. What is the greatest Corhmon measure of 720, 336 and 1736 ? ' Ans, 8- Problem lu J^ Jind the least common multiple of two or more numbers. . ' RULE.* ' ■ ■ . . ■«.•.. 1. Divide by any number, that will divide two or more of the given numbers without a remainder, and set the quotients, together with the undivided numbers, in a line beneath. 2. Divide the second line as before, and so on, till there are no two numbers that can be divided ; then the con- dnued product of the divisors and quotients will give the •multiple required. EXAMPLES. t. What is the least common multiple of 3, 5, 8 and 16? 5)3 5 8 10 _ 2)3 I § 2 3141 5X2X3X4=120 the answer. 2. Whaf ■ III — ' • " I — BTcs the less, ttrhich is absurd. Therefore 54 is the greatest com- moD measure. In the very same manner, the demonstratioa may be applied to 3 or more numbers. * The reason of this rule may also be shewn from the first ex- ample^thus: it is evident, that .3x5x8x10=1200 may be (& Tided by 3, 5> 8, and 10, without a remainder ; but 10 is a mul- ' tiplc of 5, therefore 3x5x8X2^ or 240, is also divisible by 3, 5, 8, and 10. Also' 8 is a multiple of 2 ; therefor f? 3 X 5 X4X as 120 is also divisible by 3, 5, 8, and 10 ; and ' evidently the least Aumber that can be so divided. REDACTION 0* VULGAH FRACTIONS. 6^ % What is the least common multiple of 4-and ^ ? Ans. 12. 3. What Is the least number, that 3, 4, 8 and 12 will {neasure ? - Ans. 24. 4, What IS the least number that can be divided by the ;:iine digits, without a remainder ? Ans. .2520- NS. Cr\: Reduction of Vulgar Fractio. Reduction of Vulgar Fractions is the bringing them out of one form into another, in order to prepare them for the operations of addition, subtraction, &c. CASE I. To abbreviate or reduce fractions to their lowest terms. RULE.* Divide the terms of -the given fraction by any number that will divide th^m without a remainder, and these quo- tients I • -* That dividing both the terms of the fraction equally, by any ' number whatever, will give another fraction equal to the former, is evident. And if those divisions are performed as often as can bf donCy or the common divisor be the greatest possible, the tenns of the resulting fraction must be the least possible. Note i . Any number ending with an even number, or a cy- pher, is divisible by 2. f. Any number ending with 5, or o, is. divisible by 5. 3. If the right-hand place of any number be o, the whole is divisible by 10. 4. If the two right-hand figures of any number are divisible by ^, the whole is divisible by 4. 5. If the three right-hand figures of any number are divisible by 8, tlie whole is divisible by 8. 0, If the sum of die digits constituting any number be divisi- ble by 3, or 9, the wl^ole is divisible by 3, or 9. ■ . " 7. If 6i^ JtRITHMfiTIC* tients «gain in the same manner ; and so on^ till it :lppeaxs that there is no number greater than i, which will divide thcmj and the fraction wiU be in its lowest terms. Divide both the tcrins of the fraction by their greatest common measure, and the quotients will be the terms of the fraction required. EXAMPLES. 1. Reduce |^ tO' its lowest terms. (») (») (3i (») (») i^=Tro=l|55HWl%i5|itheanswer, Or thus : I44)240{t 144 96 48)96(t Therefore 48 is the greatest common measure^ an4 48)t4o-^t> ^^ ^^^^ *s before. 2. Reduce 7. All prime nainbers> except 2 andj, have i, 3, 7> or 9, in the place of units ; and all other numbers are composite. 8. When numbers^ with the sign of addition or subtraction be- tween them, are to be divided by any number, each of the num- bers must be divided. Thus ijLJZlI5= 2 4-44-5=: 1 1. 2 9. But if the numbers have the sign of multiplication between them, only one of them m«st be divided. Thus 1^ ^ — „ 3 X4X io _ I X4X 'Q ^ I X 2 X 10 ,,20=;20. ILEbWCTION OF VULGAR FRilCTIONS. 6j tt. Reduce -^y^ to its least term^ Ans* -^ . 3« Reduce —^ to its lowest terms. Ans. y. 4* Bring ^Ti to its barest terms. Ans. ^ t. Rfdiice y|4 to its least terms. Ans. -^ 6. Reduce j^f ^ to its least terms. Ans. -^ - 7. Reduce -rfir *o »^s lowest terms. Ans. is 8* Abbreviate jV*V9V/o ^ Q^^ch as possible. Ans. ..V^ijVr CASE II. Ji redmt' a mixed f7 umber to Its equivalent imprcfer fracticn. RULE.* Multiplf the whole number by the denominator of tlic fraction, and add the numerator to the product, then that sum written above the denominator will form the fraction Required. tXAMPlES. |. Reduce 2 7 J- to its equivalent improper frnctii)n. 9 ^43 2 ^245 9 ^.27X9±i=,i^ the answer. 9 2. Reduce * All frattions represent a division of the numerator by the dc. nomiDato>r, and are taken altogether as proper and adequate ex- pressions for the quotient. Thus the quotient of 2 divided by 3 is J ; from whence the rule is manifest ; for if any number is raul- ti[.lied and divided by the same nambery it is evident tlic quotic.*:^ masi be the same as the quantity first given* I 66 ARITHMETJPC. - 2. Reduce 133-2^ to its equivalent" improper ft-Actioff. 3. Reduce 5147^ to an Improper fraction^ ' Ans. -^5-. .4. Reduce ioo4|- to an improper fractioa. Ans. -^-* 5. Reduce 47-|-:5;|-|. to an improper fraction. Ans. ^|^^*-, '•'■''-• CASE ttU Sri reduce an improper fraction to its eaulvalent *ivho!e or n::x^ ed ftutnbef. RUL£.* ■ Divide the numerator by the denominator, and the quo- tient vi^ill be the whole or mixed munber required, j; EXAMPLES. i. Reduce ^-~ to its equivalent whole or mixed numbeA i6)98i(6i-iV 5 Or, ^~y zr98i-M6=6i-y^ the answer. 2. Reduce •'- to its equivalent whole or mixed number. An3. 7. 3. Reduce ^-^ to its equivalent whole or mixed num- • ber. Ans. $6^^^. .4. Reduce ---■-^- to its equivalent whole or mixed huit>- bcr. Ans. 183^^-. 5. Reduce — /f:p- to its equivalent whole or mixed number. Ans. xaop—^- . CASE. iF' ' . '■ ' ^-^ * This rule is plainly the reverse of the former, and has its reacon in the nature of common division,' REDUCTION or VULGAR FRACTIONS. 6j CASE IV. 21; reduce a 'whole number to an equivalent fractvjTJy having a given denominator^ • ^ULE.* I^Iwitiply the whole number by the given denominator, and place tlie product over the said denominator, and it Ivill form the fraction required. 3EXAMPLE3. «, Reduce 7 to a fraction, whose denominator shall be o. 7X911:63, and -^ tlu! an>5W?r. And y =(^3—9:^:7 the prcof. ?. Reduce 13 to a fraction, whose dtriOminatcr shall be 12. Ans. y^^ 3. Reduce 100 to a fraction, whose dcnominaicr j:hall be 90. Aho. — ^~-- CASE V. 21? reduce a compound fraction to on e^uivcikni single cv.e. RULE.f Multiply all the numerators together for the numcr.itOT-, and all the denomin?.tors together for the ceur.Tninatcr, and they will form the single fraction iccuifcd. If * MidtipUcation and division are here ccually used, and coni:e- ^uently the result is the same as the (quantity first proposed. f That a compound fraction may be represented by a single oae is very evident, since a part of a psrt nr.ust be equal to some part of the whole. The truth of the rule for this rcdiiciion niny be shewn as follov/s. Let the compound fraction to be reduced be \- of %* Tb.cn -f of 4-=T-^3=TT> and consequently 4 of 4=-/,-X ? = rV ^lie sam/^ as by the rule, and the like will be found to be true iu all cas.r. 68 A1UTHMETXCU If part of the compound fraction be a whole or mixe4 number, it must be reduced to a fraction by one of the for-? mcr cases. When it can be done, divide any two terms of the frac- tion by the same number, and use the quotients instead thereof. EXAMPLES. 1. Reduce •!• of I- of -^ of ^ to a single fraction. 2x3x4x8 ^192^ x6^j^^ ^^^,^^ \ 3x4x5x11 660 55 Or, ^X*X5Xli 2. Reduce y of -f- of -|- to a single fraction. ' Ans. ^ 3. Reduce -| of j of |- of -j^ to a single iraction. Ans. -3^ 4. Reduce -j-— of -r^ of -p^ of 10 to a single fraction. ' Ans. Vi^'^ CASE VU iTo reduce fractions of different denominators to equivalent fractions^ having a common denominator^ ]^ULE I.* ^ Multiply each numerator into all the denominators, ci^^ cept its own, for a new numerator 5 and all the deiiomi-, nators continually for the common denoiiiiinator. EXAMPLES. If the componDd fraction consist of more numbers than 2, the two first may be reduced to one, and that one and the third wil^ be the same as a fraction of two. numbers ; and so on. * By placing the numbers multiplied properly under one anotlicii it will be seen, that the numerator and denominator of every frac-. tioq toliUCTlON Oir TULGiR FRACTIONS. 69 EXAMPLES. 1. Reduce -{'J -|- and -\, to equivalciit fractions, having 9 common denominator. I X 5 X 7 = 35 the new numerator for ^.. 3X2X7=42 do. for |- 4X2X5=40 do. fori. 2X5X7 = 70 Uic common denominator. Therefore tiie new equivalent fractions are -|"|-, —5-, an4 ^£-, the answer. 2. Reduce 4-> -yj "I- ai'^d -^, to fractions, having a com- mon denominator. Ans. -[-i-f , i-ff , fj-^, ^^. 3. Reduce ~, -l of j, 5-^ and yV> ^^ ^ common denonn* 4» Reduce -j j» 4- of ij, yV ami 4- to a cummon dcr nominator. Ans. vH-f^ T^'^'-h -i-^- --1 -?-^4tx- |IULE 1^; JV reduce any given fractions ij others ^ which shall have ili least common denomwator* 1. Find the least common multiple of all the denomina- tors of the given fraqtions, and it will be the comirxn dcr nominator required. 2. Divide the common denominator by the denomina- tor of each fraction, and multiply the quotient by the nu- incrator, and the product will be the numerator of the fraction required. irXAMPLES. tion are multiplied by the very same number, and consequently their values are not altered. Thus in the fust example : X5X7 3 X5X7 5 X2X7 X2X5 X2X5 ^ X2X7 7 In the 2d rule, the common denominator is a multiple of aU the dcnoniinators, and consequently will divide by any of thcr.i j it is manifest, therefore, that proper parts may be taken tor a}l tl*c ^mmerators as required. -70 AWTHMETie. EXAMPLES. I. Reduce -f, j- and |, to fractions, having the least common denominator possible. 3 6 I X I X I X 2 X 3=6 least common denominator. ,6-r2Xi=3 the first numerator; 6-^3x2=4 the second numerator; 6-^6x5:=:5 the third numerator. Whence the required fractions are |-, -^y -^ 2. Reduce -^-^ and -—g- to fractions, having the least com- mon denominator. Ans. 4-4., iLl. 3. Reduce |-, y, ^ and |-, to the least common denomi-p nator. ^ Ans. ^, -V. rT» ^t- 4. Rednce y, |-, -|. and •/— , to the least common cienom- inatcr. Ans. i-i, -^^, i-^. -^ 5. Reduce y, -J, -J-, |-, -f^ and ^^^ to equivalent frac- tfonsj having the least common denominator. Ana «^ 3tf AO 4 % 33 34 ^"S. 4yj.-4g-, Zf^y "4g-, 4g^ 4^. ^ CASE VII. 7oJind the value of a % ft- action in the known parts of the integer* KULE.* Multiply the numerator by the parts in the next inferior denomination, and divide the product by the denominator ; and if any thing remain, multiply it by the next inferior denomination, and divide by the denominator as before ; and so on as far as necessary ; and the quotients placed af- ter one another, in their order, will be the answer required. . EXAMPLES. * The numeratgur' of $. fraction may be considered as a remain- der, awl the d^i|Mliiator as a divisor^;^ therefore this rule has its rejaon in the natSre of coimpoend division. / REDUCTION OF VULCAU FRACTIONS. 'Jl EXAMPLES. % What IS tlie value of ~ of a shilling ? 5 12 7)^0^83. 2?xl. Anj. 4 4 14 2 2. What is the value of -^- of a pound sterling .^ ■■ \ Ans. 7S» 6fi. 3^-. What is the value of -f of a pound Troy ? , Ans. 70Z. 4dwt* 4. What is the value of ~ of a pound avoirdupois ? • Ans. 90Z. 2-jdi\ 5. What i.^ the value of ~ of a cwt. ? Ans. 3qr8. 31b. ijPbz. i2-|^Jr. 6. What IS the value of ~ of a mile ? Ans. ifur. i6pls. 2ydfi. ift. p^^im- 7. What is the value of -^ of an ell English ? Ans. aqrs. 3~nl3. 8. What is the value of |- of a tun of wine ? Ans. 3hhd. 3lgal. 2qts. 9. WKat 13 tlic value of ■— ^^ ^ ^^7 ^ Ans. i2h. 55' 23-rj.", CASE Vlil. ?j reduce a fraction cf cm demminathn to that cf another^ rf- iaining the same value, RULE.* Make a compound fraction of it, arid redtice it to a sin- gle QSkCf. , . :^ - BXAMFLE9. * The reason of this practice is explained ill tM mlc tog tp- ducinz conipoand fractions to single ones. v 72 ARITHMETIC; EXAMPLES. 1. Reduce -f of a penny to the fraction of a pounA I- of -V of ^rr-p-^n-i- the answch And^ of y of V±:^f=id. the proof. 2. Reduce - of a farthing ^o the fraction of a pound. 3. Reduce -rV^' ^^ the fraction of a penny. Ans. ^ •' 4. Reduce — of a dwt. to the fraction of a pound Troy. Ans. -3.;.^ 5. Reduce 4 of a pound avoirdupois to the ffaction of a cwt. Ans. --|-^. 6. Reduce ^-^V^ ^^ ^ ^'^^* ^^ ^^^^ t<^ the fraction of a pint. Ans..-^y. 7^ RedtKce -^-^ of a month ta the fraction of :a day. 8. *Reduce 75. 3d. to the fraction" of a.pound. Ans.-f-" 9. Express 6fur. i6pls. in the fraction of a mile. Ans.y.^ Addition of Vulgar Fp^actions^ Reduce compound fractions to single ones ; mixed * numbers to improper fractions ; fractions of different inte- gers The rule miglit have been distributed into two or three difFerent cases, but the directions here given may very easily be applied tcf any question^ that can be proposed in those cases, and will be more easily understood by an example or two, than by a multiplicity (#f trords. * Thus 7s. 3d. 2= 87d. and il.=:24od. r.^z^zn^^ the ans^ver'. t Fractions, before they are reduced to a common denomina- tor, are entirely dissimilar, and therefore cannot be incorporated with one another ; but when they*afe reduced to a common de* aominatori atul made parts of thc^ same tiling, their sum or differ* encet ^ iDDlTlON OF VULGAR FRACTIONS. 73 srcrs to these of tlie same ; and all of them to a common 'icnominator ; then the sum of the numerators, written over die common denominator, will be tlie sum of tlic fractions required. EXAMPLES. 1. Add 3^, "I", \ of \ and 7, together. Then the fractions are ^ , |-, ^q ^^^ T > •"• a$x8xiox 1=2320 7X8XIOX 1= 560 7X8X 8X 1= 448 7X8X 8X10=4480 7808 =: 1 2 J4I = 1 2f the answer. 8:k8XioX 1=2 640. 2. Add ^, 7~ and \ of -\ together. Ans. 8^-. 3. What is the sum of -|-, y of y and 9^ ? Ans. lo—^-. 4. What is the sum of To ^^ ^* "t" ^^ T ^^^ 7t ? Ans T^'<>P 5. Add yi. -^s. and -/i;- of a penny together. Ans. iHf , or 3s. id. iffp. 6. What IS the sum of -y of iS^ Syh -f- of -f of I- ol a pound and y of ^ of a shilling ? Ans. 7I. 1 7s. 5 i-d- 7. Add y of a yard, -^ of a foot and •§• of a mile to-., gether. Ans. 66oyds. 2ft. pin. 8. Add y of a week, -J of a day and -j- of an hour to- , gether- Ans. 2d. i44-h. '^' SUBTRACrJOI^ ence, may then be as properly expressed, by the sum or difference of the numerators, as the sum or diflference of any two quantities whatever, by the sum or difference of their individuals ; whence the reason of the rules, both for additioiv. and subtraction, is tranirpi:, K 74 -ARITHMETIC. Subtraction of Vulgar Fractiotsis, RULE. Aprepare the fractions a& in addition, and the difference of the numerators, written above the common denomina- tor, will give the difference of tlie fractions required. I ESlAMPLES. From I take ^ of \. J of f=:, and j=z ' 5 ..14 »— X«^4 the anwser required. •ai »i zi 7 2. From ^V take \. Ans. -HI. 3. From 96y take i/\\. Ans. 8iff- 4. From 14-^ take yof 19* • Ans. lyV 5. From ~1. take -^^s. Ans. 9s. 3d, 6. From ^z. take |<lwt. • Ans. iidwt. 3gr. 7/ From 7 weeks take 9^ days. Ans. 5w. 4d. 7I1. 1 2'"* Multiplication of Vulgar Fractions. "rule.* Reduce compound fractions to single ones, and mixed numbers to improper fractions ; then the product of the numerators is the numerator ; and the product of the de^ nominators, the denominator of tfie product required. EXAMPLE! % . * Multiplication by a fraction implies the taking some part ot parts of the multiplicand, and, therefore, may be truly expressed by a compound fa[B^n. T/4H|t^multiplied by | is the same as i of i ; and «^||ipttNsctions.Q^^ agree with the metliod al- ready g^en tb'^MbtSe these fS^ms to single ones, it is shewn to beiright. ■•■.'J Ans. 1 1 8' Ans. tV- Ans. »i Ans. DIVISION OF VULCAR FRACTIONS* 75 EXAMPLES. I. Required the continued product of 24-, -g-j -5- of |- and 2. ' 2irzh i of i = iii|=TV» and 2= J ; 3X6 •ften i XiX^Vx4= i^-g X /g x^"''" *' '^''^''' 2. Multiply -jSj by •^. 3. Multiply 4^ by I- 4. Multiply 4- of 7 by -|-. J. Multiply I of I by I of 3^. 6. Multiply 4y, "1: of -y and i8j, continually together. Dins ION of Vulgar Fractions. RULE.* Prepare tlie fractions as in multiplication ; then invert I die divisor, ^nd proceed exactly as in multiplication. EXAMPLES. ( J. Divide y of 19 by j of -I- 2 I ••• YXt=-7TT^ = V=7t ^e quotient required. 2. Divide * The reason of the rule may be shewn thus : Suppose it were Wl?equired to divide |- by f. Now ^-t-2 is manifestly f of ^.br — ^ $ but yizf of 2, ,'. y of 2, or f must be contained 5 timej 4X2 as often in 4^ as 2 iz : that is ?.-— i = the answer ; which is 4X2 cording to the rule ; and will be so in all cases. Note. — A fraction is multiplied by an integer, by dividing the denominator by it, or multiplying tlie numerator. And -divided by an integer, by dividing the numerator, or multiplying the '4?- pominator. I 76 4RITHldBmc. 2. Divide ^ by y. ^ Ans. ^ 3. Divide 9^ by -f of 7. - Ans. 2^ 4. Divide 3^ by 9^. Ans. f* 5. Divide |. by 4. Ans. -^ 6f Divide i of |. by -|. of |^ Aps. j. DECIMAL FRACTIONS, ■>.-.. L ' -i » A. DECIMAL is a fraction, whose denominator is an unit, or i, with as many cyphers annexed as the numera- tor has places ; and is commonly expressed by writing the ountf rator only, with a point before it called the septiratri:;% ,^t Thug, 0*5 is equsj to 4^ or 4. ^'15 -^ or ». 13 t4 or It^. 24-(J 24t^. •02 ,pj^ or /^. •0015 TTnnnr or .3^^ A jff»/V^ decimal is that, which ends at a certain number of places. But an infinite decimal is that, which is under** ' stood to be indefinitely continued. ^t'-- A repeating decimal has one figure, or several figures, • continually repeated, as far as it is found. As '33, &c. '-■•'■" . . • • • • which is 2l single repetend. And 20*2424, &c. or 20*246246, &c. which are ^Mif^2^/z//r^£fMia|/, Repeating decimals are al- so called circul4Us,OT circulating decimals. A point is set over ^ ^C**j^ a single DECIMAL FRACTIONS. 77 a single repetend, and a point over the first and last figuics of a compound rcpetend. The first place, next after the decimal mark, is loth parts, the second is 100th parts, the third is 1 000th parts, and so on, decreasing toward the right by icths, or in* creasing toward the left by icths, the same as whole or integral numbers do. As in t!ie following SCALE OS NQTAriOW. rs CO P- • CO "^ C »3 rt *3 M <o re CL, 13 5 ti "^ c . rt v> CO =3 -T? c y^ ck ^*3 3. vm •5 CO CO i « ^ s Si .2 -a <o 3 -0 Mill Hun 1 8 8 8 8 8 ? •? c S o c CO rC ^ S -S -S O *-• -^^ ?^ S -. ri '.IS 8 8*88888 8 Cyphers on the right of decimals do not alter their value. For '5 or 7^ is 4. And -50 or ^V^ is f And -500 or -ji^ is ^. But cyphers before decimal figures, and after the separ- ating point, diminish the value in a tenfold proportion for every cypher. So '5 is But -oj is And '005 is And so on. So that, in any mixed or fractional number, if the separ- ating point be moved one, two, three, &c. places to the right hand, every figure will be 10, 100, icoo, &c. times greater than before. But if the point be moved toward the left hand, then jBvery figure will be diminished in the same manner, or the whole quantity will be divided by xo, 100, xooo, 8cc. Ai>JOirjoN tV or I T f off or z T6 6"6 or TO 4' 7t A&ITRUBTIC. Addition of Decimal's. T. Set the numbers under each other according to the value of their places^ as in whole numbers^ or so that the decimal points may stand each directly under the pre? ceding, v. 2. Then add as in whole numbers^ placing the decimal point, in the^uoi, directly under the other poii^t*. EXAMPLES. (0 753^ l6'20I 3-0142 957-13 6-72819 •03014 85i3*io353 2. What is the sum of 276, 39'2i3, 7^014-9, 417, 5032 and 2214*298 ? Ans. 79993*41 1. 3. What is the sum of •014, -9816, -32, '15914, 72913 and '0047 ? Ans. 2-20857. 4. What is the sum of 27*148, 9; 8*73, 14016, 294304, 7138 and 221*7 ? Ans. 309488*2918. 5. Required the sum of 312*984, 2r39i8, 2700-42, 3'i53, 27*2 and 581*06. Ans. 3646-2088. Subtraction of Decimals, \ RULE. .1. Set the less number under the greater in the same manner as in addition. 2. Then subtract as in whole numbers, and place the decimal point in the remainder directly under the other points. , EXAMPLES. MULTIPLtCATlOM OF DECIMALS. f^ EXAMPLES. (0 214-81 4-90142 209-90858 2. From "9173 subtract* 2138. Ans. '7035. 3. From 2'73 subtract i'9i85. Ans. 0-8115. 4. What Is the difference between 91*7 13 and 407 ? Ans. 315-287. 5. What IS the difference between 16-37 and 800*135 ? Ans. 783*765. Multiplication of Decimals^ ruleA 1. Set down the factors under each other, and multiply them as in whole numbers. 2. And from the product, toward the right hand, /point off'as many figures for decimals, as there are decimal placc<% in both the factors^ But if there be not so many figures in the product as there ought to be decimals, prefix the proper number of cyphers to supply the defect. EXAMPLE?. * To prove the truth of the rule; let '9776 and -823 be tlic numbers to be multiplied ; now these are eqmvalent to rtsV?^ ^^^ 4^^ ; whence ^^^Xt'^V=i^'i^'^= -8045 648 by the na- ture of notation, and consisting of as many places, as there arc cyphers, that is, of as many places as are in both the numbers ; a&d the same is true of any two numbers whatever. r ««r ARITHMETIC* ; EXAMPLES^ (1) 91-78 •38 » 9178 7342* ^7534 ' 34-96818 *- 2- What is the product of 520*3 and '417 ? Ans. 2 1 6*9^5 f. 3. What 15 the product of 5i'6an€l 21 ? Ans. io83'6, 4. What is the product of '217 and '0431 ? Ans. '0093527. 5. What is the product of 'oji and '009! ? Ans. '0004641* Note. When decimals are to be multipfied ty ic, or loc, or 1000, &c- thait 1$, by i with any number of cyphers, it is done by only moving the decimal point so many places fur- ther to the right hand, as there are' cyphers in the said mul- tiplier ; subjoini];ig cyphers if there be not so many figures. EXAMPLES. 1. The product of 51*3 and 10 is 513* 2. The product of 2*714 and 100 is 27i'4. 3. The product of '9163 and 1000 is pi^'j* 4.. The product of 21*31 and loooo is 2i3ioo« CONTRACTION. WTfen the product would contain several more decmah than are fuccessary for the purpose in hand^ the work may be much con* tractedy and only the proper number of decimals retained. RULE. J. Set the unit figure of the multiplier under such deci- mal place of the multiplicand as you intend the last of your product MULTIPLICATION OF PfiCIMALS. 8 1 product shall bej writing the other figures of the multiplier in an inverted order. 2. Then» in multiplyingi reject all the figures in the multiplicand, which are on the right of the figure you are multiplying by ; setting down the products so that their right-hand figures may fall each in a straight line under the preceding ; and carrying to such right-hand figures from ihe product of the two preceding figures in the multipli- cand thus^ viz. X from 5 to 145 2 from 15 to 24, 3 from 25 to 34> Sec. inclusively *, and the sum of the lines will be the product to the number of decimals required, and • will commonly be the nearest unit in the last figure. EXAMPLES. t. Multiply 27* 14986 by 92*41035, so as to retain Qi^y {oar places of decimals in tlie product. Contracted. Common way. 27*14986 27-14986 53014-29 92-4'035 34434874 '3 . 57493^ 542997 81 44958 108599 2714 9«6 2715 108599 44 81 542997 z 14 24434874 2508*9280 25o3'928o 650510 2. Multiply 480*14936 by 2*72416, retaining four deci- mals in the product. Ans. 1 308*0036. 3. Multiply 73'8429753 by 4-628754, retaining five decimals in the product. Ans. 341*80097. 4. Multiply 8634*875 by 843*7527, retaining only the integers in the product. Ans. 7285699, ' L ia AHITHMETIC. DirisiON of Decimals. RULE.* Divide as in whole numbers ; and to know how many decimals to point oflF in the quotient, observe the following ruks : X. There must be as many decimals in the dividend, as in both the ^visor and (quotient ; therefore point off for decimals in the quotient so many figures, as the decimal places in the dividend eicceed those in the divisor. 2. If the figures in the quotient are not so many as the rule requires, supply the defect by prefixing cyphers. 3. If the decimal places in the divisor be more thaa those in the dividend, add cyphers as decimals to the divi- dend, till the number of decimals in the dividend be equal to those in the divisor, and the quotient will be integers , till all these decimals are used. And, in case of a remain* der, after all the figures of the dividend are used, and more figures are wanted in the quotient, annex cyphers to the remainder, to continue the division as far as necessary. 4. The first figure of the quotient will possess the same place of integers or decimals, as that figure of the divi- dend, which stands over the units place of the first product. EXAMPLESf I. Divide 3424*6056 by 43*6. 43-6)3424-6os6(78-546 305^ 3726 3488 2380 2180 2005 1744 2616 2616 2. Divide * The reason of pointing off as piatry decimal places in the quodent, as those in the dividend exceed those in the divisor, will easily OIVISIOU OF DECIMALS. 83 2. Divide JS77875 by '675. Ans. 5745000. 3. Divide -0081892 by •347, Ans. '0236. 4. piYi4e 7-13 by -18. * Ans. 39. CONTRACTIONS. L If the divisoif he an integer vnii any number of cyphers at the end ; cut them off, and remove the decimal point in the dividend so many places further to the left, as there virere cyphers cut off, pre^xing cyphers^ if need be } then pro* ceed as before. EXAMPLES. 1. Divide 953 by 21000. 21*000) 7)-3i7(J6 •04538, &c. flm I first divide by 3, and then by 7, because 3 times 7 Is si. 2. Divide 41020 by 32000. Ans. i*28i875. KoTS. Henpe, if the divisor be x with cyphers, the quotient will be the same figures with the dividend, having the deci^ mal point so many places further to the Icft^ as there ^e pjphers in the divisor. EXAMPLES, 217*3 -r 100 =:2'i73. 419 by 10 = 41-9. 5*i6 by 1000 = '005 16. '21 b)j 1000= '00021. |L When the number of figures in the divisor is great, the op^ eration may be contractedx and the necessary number of deci^ tnal places obtained* RULE. 1. Having, by the 4th general rule, found what place of decimals or integers the first figure of the quotient will pos-» sess '9 easily appear ; for since the number of decimal places In the divi- dend is equal to tliose in the divisor and quotient, taken togethci:, by the nature of multiplication ; it follows, that the quotient cca- tai^s as jnany as the dividend exceeds the divisox;. \ 84 AlUTRMfeTie. sess ; c^msider how many figures of the quotient wtll serve the present purpose ; then take the same number of the left-hand figures of the divisor, and as many of the dividend figures as will contain them (less than ten times) i by these find the first figure of the quotient. 2. And for each following figure, divide the last remain- der by the divisor, wanting one figure to the right more than before, but observing what must be carried to the first product for such omitted figures, as in the contraction of Multiplication i and continue the operation till the divisor be exhausted* 3. When there are not so many figures in the divisor, as are required to be in the qtiotient, begin the division with all the figures as usual, and continue it till the number of figures in the divisor and those remaining to be found ia the quotient be equal ; after which use the contraction. E3(AMPtES. !• Divide 2So8'928otf505i by 92'4i03j;, «o as to have four decimals in the quotient.^^-In this case, the quotient will contain six figures. Hence . Contraction. 92-4X«>3i5)^5^»'928*o6505i(a7'i498 • • • t^ 1848207 *- 660721 646872 Common REDUCTION OF DECIMALS. CommoQ Way. 92 41035)2508-92806505 i(27'l49$ .I848207I •s 660721 646872 06 45 13848 9241 615 035 4607 3696 911 831 79 73' 5800 4140 16605 472901 928280 5544621 2f Divide 75n*i7562by a'257432, so that the quotient may contain three decimals. Ans. 319-467. 3. Divide 1 2' 169825 by 3'i4i59, so that the quotient may contain five decimals. Ans. 3*87377* 4. Divide 87*076326 by 9*365407, and let the quotient contain seven decimals. Ans. 9*2976554^ Reduction of Decimals* CASE I. 31? reduce a vulgar fraction to its . juivalent dcchnat RULE.* Divide the numerator by the d ►nominator, annexing a» jnany cyphers as are necessary ; aad the quotient will be the decimal required. EXAMPLE^- * Let the given vulgar fraction, whose decimal expression is reqiured, be Vr* Now since every decimal fraction has 10, ioo» 1000, 8cc. for its denominator ; and, if iwo iBractions be equal* k ^ 96 4&ITHMETIC, EXAMPLES. 1:. Reduce -5^ to a decimal. 4)5*000000 6)1*250000 •208333, &c. 2. Require^ the equivalent decimal expressions for ^^ f and ^. Ans. -25, '5 and -75. 3. What is the decimal of I- f Ans. '375. 4. What is the decimal of ^ ? Ans. '04. 5. What is the decimal of y^ ? Ans. '915625. 6. Express -5^^^ decimally. ^ns. '07^577, &c. CASE 11. 9fl reduce, numbfrs of different denomnations to their equivalent decimal values. R U L E.f 1. Write the given numbers perpendicularly under each other for dividends^ proceeding orderly from the least tc^ the greatest. 2. Opposite to each dividend, T)n the left hand, place such a number for a divisor, as will bring it to the nesjt superior name, and draw a line between them. 3. Begi^ t • " ' " " "" it will be, as ine denominator of one is to its numerator, so is the denominator of the ottr to its numerator; therefore 13:7: : looo^ '<cc. : 7Xiooo,&c.^*-^oooo,Jcc. ^ .^^g^^^ ^^ n^mtvzioT of the decimal required ; and is the same as by the rule. f The reason of the ruie may be explained from the first ex- ample ; thus, three farthings is ^ of a penny, which brought to a decimal is '75 ; consequently 9|<i. may be expressed 9'75d. but 9*75 is T-J^ of a penny zn-^^ of a shilling, which brought to, a decinjal is '8125 ; and therefore 15s. 9^. may be expressed 15-81258. In like manner 15-81258. is 'ijVoV of a shilling ^ llllll of a pound =, by bringing it to a decimal, 7906251. as, by the role. REDUCTION OF DECIMALS. 87 3. Begin with the highest, aiid write the quotient of each division, as decimal parts, on the right hand of thd dividend next below it ; and so on, till they be all used, and the last quotient will be the decimal sought. EXAMPLES. I. Reduce 15s. 9^. to the decimal of a pound. 4 12 20 3' 91S 15-8125 •790625 the decimal required. 2. Reduce ps. to the decimal of a pound. Ans. '45. 3. Reduce 19s. 5~d. to the decimal of a pound- Ans. '972916. 4. Reduce looz. i8dwt i6fft4 to the decimalbf a lb. Troy. Ans. '91 1 1 II, &€• 5. Reduce 2qTS. 141b* to the decimal of a cwt. Ans. '625, &c. 6. Reduce 17yd. ift. 6in. to the decimal of a mile^ Ans. '00994318, &c. 7. Reduce 3qrs. 2nls. to the decimal of a yard. Ans. '875. 8. Reduce igal. of wine to the decimal of a hhd. Ans. '015873. 9. Reduce 3bu. ipe. to the decimal of a quarter. Ans. '40625. 10. Reduce low. 2d. to the decimal of a year. Ans. '1972602, &c CASE III. To find the decimal of any number of shillings^ pence andfar^ things^ by inspection. RULE.* Write half the greatest even number of shillings for the first decimal figure, and let the farthings itk the given pence and ^ The invendon of the rule is as follows : as shillings are so many 20ths of a pounds half of them must be so many loths, and consecjaentiy 88 ARITHMETIC. and farthings possess the second and third places ; observ- ing to increase the second place by 5, if the shillings be odd i and the third place by i, when the farthings txceed IZ ; and by a, when they exceed 37. KTAMFLES. i. Find the decimal of 15s. S^d. by inspection, i =fof 14s. 5 for the odd shilling. 34 zz farthings in S^d. I for the excess above 12» •785 — decimal required. 2, Find by inspection the decimal expression of t6s. 44^. md X3S. ic|<i. Ans. '819 and '694. 3. Value the following sums by inspection, and find their total, viz. ips. ii-Jd. -f- 6s. 2d. + ^^b. 8^. + is. lo^d. + -Jd. -f" J-Jd. Ans. 2*043 the total. CASE consequently take the place of loths in the decimal ; but when they are odd, their half will always consist of two figures, the first of which will be half the even number, next less, and tlic second a 5 ; and this confirms the rule as ^ as it respects shillings. Again, farthings are so many 96oths of a pound ; and had it happened, that 1000, instead of 960, had made a pound, it L plain any number of farthings would have made so many thou- sandths, and might have taken their place in the decimal accoid- ingly. But 960, increased by vy part of itself, is =: loco ; con- sequently any number of farthings, increased by their ^j part, will be an exact decimal expression for them. Whence, if the num- ber of farthings be more than 1 2, a iV P^'^^ is greater than i, and therefore 1 must be added ; and when the number of farthings is more than 37, a Vt pa^t is greater than i|, for which 2 must be added ; and thus the rule is shewn to be right. REDUCTION OF DECIMALS. 89 CASE IV. 91? Jlfid the value of any given decimal in terms of the integer. RULE. 1. Multiply the decimal by the number of parts in the next less denomination, and cut off as many places for a remainder on the right hand, as there are places in the giv- . en decimal, 2. Multiply the remainder by the parts in the next infe- rior denomination, and cut off for a remainder, as before. ' 3. Proceed in this manner through all the parts of the integer, and the several denominations, standing on thcleft hand, make the answer. EXAMPLES. I. Find the value of '37623 of a pound. 20 7*5 2460 12 6-29520 4 1*1 8080 Ans. 7s. 6id. 2. What is the value of '625 shilling ? Ans. 7^d- 3. What is the value of -83229161. ? Ans. i6s. 7^^. 4. What is the value of •6725cwt. ? Ans. 2qrs. 191b. 50Z. ^. What is the value of -67 of a league ? Ans. 2mls. 3pls. lyd. 3in. ib.c. 6. What is the value of '61 of a tun of wine ? Ans. 2hhd. 27gal. 2qt. ipt. 7. What is the value of •461 of ^ chaldron of coals ? Ans.^ i6bu. 2pe.' 8. What is the value of '42857 of a month ? Ans. iw. 4d. 23h. 59' 56''. CASE M JO AUITHMETIC. CA^E V. To find the value of any decimal of a pound by inspection. RULE. Double the first figure, or place of tenths, for shillings, and if the second be 5, or more than 5, reckon another shilling 5 then call the figures in the second and third places, after 5, if contained, is deducted, so many farthings ; abating i, when they are above i2 \ and 2, when above 37^ and the result is the answer. . EXAMPLES. I. Find the value of '7851. by inspection. 14s. = double 7. IS. for 5 in the place of tenths. 8i = 35 farthings. j^ for the excess of 12, abated. 15 s. B^d. the answer. ■ 2. Find the value of '875I. by inspection. Ans. 1 7s. 6d. 3. Value the following deciftials by inspection, and find their sum, viz. -9271. + '35^1- + '2031. -f '06 il. + -021. -f- '009I. - Ans. il. IIS. 5|d. FEDERAL MONET* The denominations of Federal Money^ as determined by an Act of Congress, Aug. 8, 1786, are in a decimal ratio ; and, therefore, may be properly introduced in this place. A mill. * The coins of federal money are two of gold, four of silver, and two of copper. The gold coins are called an eagle and half- eagle ; the silver, a dollar y half -dollar ^ double dime and dime ; and the copper, a cent and halfcenU The standard for gold and silver is eleven pans fine and one part alloy. The weight of fine gold FEDERAL MONET. pi A mill, which is the lowest money of account, is '00 1 of a dollar, which is the money unit. A cent is •01 Or 10 mills = I cent. A dime •I marked m. c. A doUar !• 10 cents = I dime, d. An eagle lO- 10 dimes = i dollar, D. 10 dollars = i eagle, E. A number in the eagle is 246*268 grains ; ¥)f fine silver in die dollar, 375*64 grains ; of copper in 100 cents, 2^1b. avoirdupois. The fine gold m the half-eagle is half the weight of that in the eagle ; the fine silver in the half-dollar, half the weight of that in the dollar, &c. The denominations less than a dollar are expressive of their ?aiues : thus, mlil is an abbreviation of m;7Zf, a thousand, for 1000 mills are equal to i dollar ; cenf^ oi centum j a hundred, for 100 cents are equal to i dollar ; a dime is the French of tUhey the tenth part, for 10 dimes are equal to i dollar. The mint-price of uncoined gold, 1 1 parts being fine and i part alloy, is 209 dollars, 7 dimes and 7 cents per lb. Troy weight 5 and the mint-price of uncoined silver, 1 1 parts being fine and i part alloy, is 9 dollars, 9 dimes and 2 cents per lb. Troy. In Mr. Pike's "Complete System of Arithmetic," may be seen ** Rules for reducing tlie Federal Coin, and the Currencies of the several United States ; also English, Irish, Canada, Nova- Scotia, Livres Tournois and Spanish milled dollars, each to the par of ajl the others.*' It may be sufficient here to observe re- specting the currencies of the several States, that a dollar is equal to 6s. in New-England and Virginia ; 8s. in New- York and North- Carolipa ; 7s, 6d. in New- Jersey, Pennsylvania, Delaware and Maryland ; and 4s. 8d. in South-Carolina and Georgia. The English standard fi>r gold is 22 carats of fine gold, and z carats' of copper, which is the same as 11 parts fine and i part al- loy. The English standard for silver is i8oz. 2dwt. of fine silver, and l8dwt. of copper ; so that the pit)portion of alloy in their sil- ver is less than in their gold. When either gold or silver is finer or coarser / ft IIUTHMSTIC. A number of dollars^ as 754, may be read 754 doU^ifs, or 75 eagleSy 4 dollars ; and decimal parts of a dollar, as •365, may be read 3 dimes, \6 cents, 5 mills, or 36 cents, 5 mills, or 365 mills ; and others in a similar manner. Addition J subtraetion^ ntultipUcation ?nd division of federal money are performed just as in decimal fractions •, and consequently with more ease than in any other kind of money. ^ EXAMINES. I* Add 2 doUars, 4 dimes, 6 cents, 4D. 2d., 4d. pc., lE. 3D. 5c. 7m., 3c. 9m., iD. 2d. 8c. im., and 2E. 4D, 7d. 8c. 2m. together. (i) (3) E. D. d. c. m, E. D. d. c. m, E. D. d, c. m. 2-46 34*123 30-67X 4-2 1-178 3123 •49 78-001 4*567 13-057 1*7 '03 •039 i % 70*308 I-28I 61*789 7*17 ^4 -782 6-341 8-231 46-309 Ans, From Subtract (4) (S) (6) E.D. d. cm. E.D. d.c.m. D. d.c.m. 3 2/ • I 7 8 70-000 a • 6 5 2 17-289 7*813 • 7 Remain. 14*889 7. Multiply coarser than standard, the variatioa from standard is estimated by carats and grains of a carat in gold, and by penny-weights in sil« ver. Alloy is used in gold and silver to harden them. Note. — Carat is not any certain weight or quantity, but iV of any weight or quantity ; and the minters and goldsmiths diride it into 4 equal parts, called grains of a carat. CIRCULATING DECIMALS. 931 7. Multiply 3D. 4d. 5c. im. by iD. 2d. 3c. zm. D. 3*45 1 Note. •The figures after ^ or on the right hand of, mills 6qo% ^re decimals of a mill. ^0353 6902 3451 4-251632 =4-251 AW Ans. D. D. 8. Multiply 6*347 by 4*5 3 2. D. D. 9. Multiply 7i'oi2 by 3*703. D. 10. Multiply 8o6*222-i-by 9. D. D. 11. Divide 4-251632 by 1*232. I -232)4-25 1632(3*45 1 Answer. 3696 555^ 4928 6283 6160 Ans. 28*764664. D. Ans. 262*957436. D. Ans. 7256. 1232 1232 12. Divide 20D. by 2000. 13. Divide 7256D. by 9. D. Ans. o*oi« D. Ans. 806*222-^ CIRCULATING DECIMALS. It has already been observed, that when an infinite deci- mal repeats always one figure, it is a single repetend ; and wJien more than one, a compound repetend ; also that a point is ■*■ 94* A%ITHlfETIC« is set over t single repetend^ and a point over the first and last figures of a compound repetend. It nia^be furthef obserred, that when other decimal fig- ures precede a repetend, in any number, it is called a • • • mxed repetend : as '23, or •104 123 : otherwise it is ^pure^ or • • • simple^repetcnd: 2iS*'^znA *12'^. • m Similar repetends begin at the same place : as ^3 and *6^ '■ • • • • or i'34i and 2*i$6* ' Dissimilar irepttenis begin at differeot places : as '253 aad •4752- Conterminous repetends end at \he same place : as "la^ and "009. Similar and conterminous repetends begin and end at the • • • • same place : as 2*9104 and *66i3. Reduction of Circulating Decimals. CASE i* To reduce m simple repetend to its eauivalent vulgar fraction. RULE.*' r. Make the given decimal the numerator, and let the denominator be a number consisting of as many nines as there are recurring places in the repetend. 2. If — — — — * ■ I ' ■ ■— — ■ ■ —— ^ii— ^— ^ II I ■ * If unity, with cyphers annexed, be divided by 9 ad infinitum^ the quotient will be i continually ; i. e. if -^ be reduced to a deci- maly it will prodnce the circulate' *x ; and since *i b the decimal equivalent to |, '2 will =: f , '3 = |, and so on till •9 = 1 =r i. Therefore, every single repetend is equal to a vulgar firaction, whose numerator is the repeating figure, and denominator 9. Again, RE:DUCTI0N of circulating iDSCIMiULS. 9; 2. If there be integral figures in the circulate, as many {cyphers must be annexed to the numerator, as the highest place of the repetend is distant from the decimal point, EXAMPLES. • • a 1. Required the least vulgar fractions equal to '6 and * 1 23. •6=|=f } and -Wszz^^:^ Ans. 2. Rcdi^ce '3 to its equivalent vulgar fraction. Ans. -j. 3. Reduce i'62 to its equivalent vulgar fraction. Ans. Vt? • 4* Required the least vulgar fraction equal to '769230. Ans. fj-. CASE II. To reduce a mixed repetend to its equivalent vulgar fraction. RULE.* I. To as man) nines as there are figures in the repetend,^ annex as n^any cyphers as there are finite places, for a de- nominator. 2. Multiply Again Y -^^^ or -^^ being reduced to decimals, makes *oioioi, &c. or '00 1 00 1, &c. ad infinitum =:*oi or 'ooi ; that ifi, ^ra •oi,and^^=:'ooi ; consequently ^V^'^^* ^=='°3» ^^' ^"* ^=•002, ■^^^=•003, &c. and the same wiH hold universally. * In like manner for a mixed circulate ; consider it as divisible into its finite and circulating pans, and the same principle will be seen to run throu^ them also : thus, the mixed circulate -16 is divisible into the finite decimal *i, and the repetend *o6 ; but •i=T?5,and*o6 would besrf, provide the circuktiou b^gan immc» dtately after the place of units ; but as it be^s after the place of 90 ARITHMETIC 2* Multiply the nines in the said denominator by the finite party and add the repeating decimal to the product, for the numerator. 3. If the repetend begin in some integral place, the finite value of the circulating'^^art must be added to the , finite part. EXAMPLES. 1. What IS the vulgar fraction equivalent to •138 ?* ,♦ 9X13+8=11251= numerator, and 900Z: the de» nominator; .*. 'i38zr~3-|^z:j^y the answer. 2. What is the least vulgar fraction equivalent to -53 ? Ans. -i* 3. What is the least vulgar fraction equal to '5925 ? X Ans. '"^ 4. What is the least vulgar fraction equal to '008497 133; 5. What is the finite number equivalent to 31*62 .^ Ans. 31-I4. CASE III. To make any number of dissimilar repetends similar and con^ terminous^ RULE.* Change them iflfeo other repetends, .which shall each consist of as many figures as the least common multiple of the tens, it is ^ of -n^zr^, and so the vulgar fraction =:*i6 is tV-I* ^=:/?y+^=^, and is the same as by the rule. * Any given repetend ^atcver, whether single, compound, pure or mixeda may be transformed into another repetend, that shall consist. ftfiDUCTlON OF CIRCXTLATIMG DECIMALS. 97 the several numbers of places, found in all the repetends^ contains units. StAMPLES. I. Dissimilar. Mad^ similar and conterminous. 9814 =5 9*81481481 ^'S SS 1*50000009 8726 cr Sr26666666 •083 = •08333333 124-09 as 124-09090909 ^. Make '3, *2^ and '045 similar and conterminous. •• •••■ •• ■ 3. Make '321, *8262, '05 and 'O902 similar and con- ienninous. w • • • • 4. Make '5217, 3*643 and 17' 123 similar and conter- nunous. CASE IV, To find whether the decimal fraction^ equal to a given vulgat wiey be finite or infinite^ and of how many places the repe^ lend will consist. RULE.* I. Reduce the given fraction to its least terms, and di- ride the denominator by 2, 5 or 10, as often as possible. 2. If consist of an equal' of^eater number of figiu^at pleasure : that •4 may be transformed to '44, or -444, or -44, &c. Also '57=5 •5757='5757=*575 j and so on; which is too evident to need any further demonstration. ♦ In dividing I 'oooo, &c. by any prime number whatever, e»- ' ccpt 2 or 5, the figures in the quotient will begin to repeat over sgain as soon as the remainder is i. And since 9999» &c. is less tbto N 98 ARITHMSTIC. 2. If the whole denominator vani^ in dividing by 2, 5 or I0| the decimal will be finite, and will consist of &o ma- ny places as you perform divisions. 3. If it do not so vanish, divide 9999, Ac, by the result, till nothing remain, and the number of 9s used will shew the number of places in the repetend 5 which will begin after so many places of figures, as there were los, as or 5s, used in dividing. EXAMPLES. I. Required to find M'^hether the decimal equal to -?-i-^ be finite or infinite ; and, if infinite, how many places the repetend will consist of. a » a ' ■rr5^=2|T3- I 8 I 4 I 2 h i therefore the decimal is finite, and consists of 4 places. 2. Let than 1 0000, &c. by i, therefore 9999, &c. divided by any num- ber whatever will leave o for a remainder, when the repeating fig, ures are at their period. Now whatever number of repeating figures we have, when tlie dividend is i, there will be exactly the same number, when the dividend is any other number whatever. For the product of any circulating number, by any other given num* ber, will consist of the same number of repeating figures as before. Thus, let '507650765076, &c. be a circulate, whose repeating part is 5076. Now every repetend (5076) being equally multi- plied, must produce the same product, r or though these products will consist of more places, yet the overplus in each, being alike, will be carried to the next, by which means, each product will be equally increased, and consequently every four places will con- tinue alike. And the same will hold for any other number what- ever. Now hence it appears, that the- dividend may be altered at pleasure, and the number of places in the repetend will still be the same : thus t:^=:*90, and A* or iVX3='27, where the number of places in each is alike, and the same will be true in all cases. ADDITlOirOF eXACULATlMC DECIMALS. 99 c. I-ct -pj- be the fraction proposed. 3. Let y be the fraction proposed. 4. Let ■^:^ be the fraction propose^. 5. Let -g-T^? ^ ^^^^ fraction proposed. Addition of Circulating Decimals. RULE.* X. Make the repetends similar and conterminousj and pn4 their sum as in common addition. 2. Divide this sum by as many nines as there are placet in the repetend, and the remainder is the repetend of the sum ; wbich must be set under the figures added, with cyphers on the left hand> when it has not so many places as the repetends. 3* Carry the quotient of this division to th^ next cok vxan^ and proceed with the rest as in finite decimals. EXAMPLES. • • . . • • • 1. Let 3'6+78-3476+73S-3+375+'27+i87'4l>c add, ^ together. Dissimilar^ Similar and conterminous. • • • 3'6 == 2'^^^^^^^. 78-3476 = 78-347647^ 735-3 = 735-3i33333 375- = 375'ooocooo. • • . . •27 = 0-2727272 187-4 = 187-4444444 i38o'o648i93 the sum. In tliis question, the sum of the repetends is 2648 ipt-, '?rhich, divided by 999999, gives 2 to carry, and the re- mainder is 648193. 2. Let ■■ " ■■■■ ■ ■ I I I I I !■■ I 11 ■■-.■ ■ . . ■ ■ * These rules are both evident from what lias beea said in re-. 4uctiofi. / riM ARITHUBTIC. 2. Let S39i-357+72-38+i87ai+4-2965+2i7-849^ • • • ^42'i764-'523+58-30048 be adfle4 togetker. Ans. 5974' 1 037 1. 3. Add 9-814+ i*5+87*26+'o83 + i24'09 together. Ans. 222-75572390. •• •• •• •• • • 4. Addi62 + i34*09+2*93+97*2tf+3*76923o+99'o83 • • • • +i'5+'8i4 together. A^s. 50i*6265io77. Subtraction of Circulating Decimals.. RULE. Make the repetends similar and conterminous, aQd sub- tract as iisual ; observing, that, if the repetend of the subr trahend be greater than the repetend of the minuend, then the right-hand figure of the remainder must be less by uni« ty, than it would be, if the expressions were &iite« EXAMPLES. • • • • I. From 85*62 take 1 3*76432. • • • '• 8562 = 85-62626 • • • ' • i3-7<5432 = i3"7<S43* 71-86193 the difierence. « . • 2. From 476-32 take 84-7697. Ans. 39i*5524« • • a • • • 3. From 3*8564 take '0382. Ans. 3-81. JMiULT IP LIGATION of CIRCULATING DECIMALS. RULE. I. Turn both the terms into their equivalent vulgar frac- tions, and find the product of those fractions as usual. 2. Turn \ DIVISION OF CIRCDLATIl^C DECIMALS. JO I . 2. Turn the vulgar fraction, expressing the product, in- to an equivalent decii3(ial, and it will be the product re- quired. EXAMPLES. 1. Multiply '36 by '25. 30— 5-5— TT AX 14^=^= -0929 the product. 2. Multiply 37*23 by '26. Ans. 9*928. 3. Multiply 8574-3 by 87-5. Ans. 750730*5^8. 4. Multiply 3'973 by 8. Ans. 31*791. 5. Multiply 49640-54 by '70503. Ans. 34998*4 199003# 6. Multiply 3*145 by 4'297. Ans. 13*5169533, PinsioN of Circulating Decimals. RULE. 1. Change both the divisor and dividend into their equivalent vulgar fractions, and find their quotient as usual. 2. Turn the vulgar fraction, expressing the quotient, intd its equivalent decimal, and it will be the quotient re« quired. EXAMPLES. I. Di/ide '36 by '25. -A-5-M=A^?<*T=^= ii?T= >'42292490"8577O75O(;88i tl-»e quotient.:.- . • ' . • 2. Dividft 3i9'28oo7ii2 by 764'5, Ans. '4176325. « 3. Divide I t02 AMTHMBTIC» • • • • 3. Divide 234*6 1>y 7. An9. 3or7i4285* • • • • • • 4. Divide iy5^i^9S33 ^J f^py. Ans 3-145. OF PROPORTION IN GENERAL^ JN UMBERS are compared together to discover the re* lations they have to each other. There must be two numbers to form a comparison : tlie somber which is compared, being written first, is called the antecedent ; and that to which it is compared, ihe conse- quent. Thus of these numbers 2 : 4 ::^ : 6, 2 and 3 arc called the antecedents ; and 4 and 6, the consequents. Numbers are compared to each other two diiSercnt ways :. one comparison considers the difference of the two numbers, and is czllcd arithmetical relation, the difference being some- times named the arithmetical ratio ; and the other considers their quotient, and is termed geometrical relation, and the quo- tient the geometrical ratio. So of these numbers 6 and 3, the diflFerence or arithmetical ratio is 6—^ or -? 5 and the, geometrical ratio is I* or 2. If two or more couplets of numbers have (qual ratios,^ or differences, the equality is named proportion ,• and their terms similarly posited, that is, either all the greater, or all the less, taken as antecedents, and the » rest; a^ conscr quents, are chilled proportionals. So the two couplets 2, 4, and 6, 8, taken thus, 2, 4, 6, 8, or thus, 4, 2, 8, 6, arc arithmetical proportionals ', and the two couplets 2, 4, and PBOPO&TION. 103 85 i6> taken thus, 2, 4, 8, 16^ or thus^ 4, 2, 16, 8, are ge- ometrical proportionals.*' Proportion is distinguished into continued and discontinued. If> of several couplets of proportionals written down in a series, the difference or ratio of each consequent and tli^ antecedent of the next following couplet be the same as the common difference or ratio of the couplets, the pro- I portion is said to be continued^ and the numbers themselves a series of continued arithmetical or geometrical proportionals* So 2> 4, 6» 8, form an arithmetical progression \ for 4 — 2 = 6 — 4=8 — 6 = 2 ; and 2, 4, 8, 16, a geometrical pro- gression ; for -|^=:•|=: ^-^ = 2. But if the difference or ratio of the consequent of one couplet and the antecedent of the next couplet be not the same as the common difference or ratio of the couplets^ the proportion is s^d to be discontinued. So 4, 2, 8, 6, are in discontinued arithmetical proportion; for 4—2= 8 — 6= 2, but 8*-'2 = 6 ; also 4, 2, 16, 8, are in discontinued geomet' fical proportion ; for ^=: '^ tr 2, but -/ = 8. Four numbers are directly proportional^ when the ratio of the first to the second is the same as that of the third to tlie fourth. As 2 : 4 : : 3 : 6. Four numbers are said to be recipri- ally or inversely proportional^ when the first is to the second as the fourth is to the third, and vice verSa. Thus, 2, 6, 9 and 3, are reciprocal proportionals \ 2 : 6 : : 3 : 9. . Tlirce or four numbers are said to be in harfnonical pro* i portion^ when, in the former case, the difference of the first ' and * In geometrical proportionals a colon is placed between the terms of each couplet, and a double colon between the couplets ; in arithmetical proportionals a colon may b^ turned horizontally between the terms of each couplet, and two colons written be- tween tlie couplets. Thus the above geometrical prpportionals ara written thus, 2 : 4 : : 8 : 16, and 4 : 2 : : 16 : 8 I the arithmet- icd, 2 •• 4 : : 6 •• 8, and 4 •• 2 : ; 8 •• 6. 104 AtllTHMETlC- and second is to the difFerence of the second and diird, as the first is to the third ; and, in the latter, when the dif- ference of the first and second is to the difFerence of the third and fourth as the first is to the fourth. Thus, 2, 3 and 6 ; and 3, 4, 6 and 9, are harmonical proportionals ; for 3 — 2= 1 : 6-^3 = 3 :i 2 i 65 and 4 — 3 = 1 19 — 6=3 : : 3 : 9. Of fout arithmetical proportionals the sum of the ex- 'fjk tremes is equal to the sum of the means.* Thus of 2 •. 4 : : 6 .. 8 the sum of the extremes (2-f-8)rr the sum of the means (4+6)zzio. Therefore, of three arithmet- ical proportionals, the sum of the extremes is double the mean. Of fotlr geometrical proportionals the product of the extremes is equal to the product of the means.f Thus, of 2 : 4 : t 8 : 16, the product of the extremes (2X16) is equal to the product of the means (4X 8)1=32. Therefore of three geometrical proportionals, the product of the ex- tremes is equal to th^ square df the mean. Hence it is easily seen, that either extreme of four geo- metrical proportionals is equal to-the product of the means divided « . - * Demonstration. Let the four arithmetical proportionals be Jif J5, Cy A viz. J *- B : : C » D ; then, A—BzzC—D and B+D being added to both sides of the equation, jf — B+B+l> zzzC — D+B+D ; that is, -^+-D the sum of the extremes =:C+^ the sum p£ the means. — And three ^, B9 C, may be \ thus expressed, j1 -. B : : B *• C ; therefore jI+CiiiB+BzzzB, , Q^ E. D. j^ f Demonstration. Let the proportion ht A : S :i C i Dy A C and let ^=5=r ; then Az=:Br, and C^Dr ; multiply the for- mer of these equations by Z), and the htter by B 5 then AD:rz ' BrDy and CBzzDrBy and consequently AD the product of the extremes is equal to BC the product of the means. — And three may be thus expressed, A : B : : B : C, therefore A CzzB^ B=zB\ Q^E. D. PROPORTION. 105 divided by the other extreme •, and that either mean is equd to the product of the extremes divided by the other mean; SIMPLE PROPORTION, or RULE OF I THREE. The Rule of Three' is that, by which a number is found, having to a given number the same ratio, which is between . two other given numbers. For this reason it is sometimes ' ttuned the Rule of Proportion, It is called the Rule of Three^ because in each of its questions there are given three numbers at least. And be- cause of its excellent and extensive use, it is often named tKe Golden Rule^ . RULE.* ;, \v ; ■ down the number, which is of the same kind g'.. 'nt^f^ic .'^<jwer or number required.' ^ * ! - 0. Consider ► * Demonstration. The following observations, taken col- fcctively, form a demonstration of the rule, and of the reductions Aieationed in the notes subsequent to it. i. There can be comparison or ratio between two numbers, onfy irhen they are considered either abstractly, or as applied to things of* the same kind, so that one can, in a proper sense, be contained in the other. Thus there can be no comparison between 2^ men and 4 days ; but there may be between 2 and 4, and between 2 days Ind 4 days, or i men an^d 4 men. Therefore, the 2 of the 3 given numbers, that are of the same kind, that is, the first and thh-d, when they are stated according to the rule, are to be com- pared together, and their ratio is equal to that^ required between the remaini^ or second number and the fourth or answer. ^ ^ ^. Though O 10^ ARITHMETIC. • 2. Consider whether the answer ought to be greater or less than this number : if greater, write tjhe greater of the two remaining numbers on the right of it for the third, and 2. Though numbers of the same kind, being either of the same or of different denominations, have a real ratio, yet this ratio is the same as that of the two numbers taken abstractly, only when they are of the same denomination. Thus the ratio of il. to 2I. is the same as that of i^ to 2 =i ; i^. has a real ratio to 2I. but it is not the ratio of i to 2 ; it is the ratio of is. to 40s. that is, of I to 40 =^'o« Therefore, as the first and third numbers have the ratio, that is required between tlie second and answer, they must, if not of the same denomination, be reduced to it ; and then their ratio is that of the abstract numbers. 3. The product of the extremes of four geometrical propor- tionals is equal to the product of the means ; hence, if the pro- duct of two numbers be equal to the product of two other num- bers, the four numbers are proportionals ; and if the product of two numbers be divided by a third, Jthe quotient will be a fourth proportional to those three numbers. Now as the question is i-c > solvable into this, viz. to find a number of the same kind as the second in the statement, and having the same ratio to it, that the greater of the other two has to the less, or the less has to the great- er ; and as these two, being of the same denomination, may be con- sidered as abstract numbers ; it plainly follows, that the fourth number or answer is truly found by multiplying the second by one of the other two, and dividing the product by that, which remains. 4. It is "very evident, that, if the answer must be greater than the second number, the greater of the other two numbers must be the multiplier, and may occupy the third place ; but, if lessy the less number must be the multiplier. 5. The reduction of the second number is only performed for convenience in the subsequent multiplication and division, and not to produce an abstract number. The reason of the reduction of the PROPORTION.^ IC7 .i^ndthc other on the left for the first number or term ; but if less, write the less of the two remaining numbers in the third place^ and the other in the first. 3. Multiply the quotient, of the remainder after division, and of the product of the second and third terms, when it cannot be divided by the I £rst, is obvioiis. 6. If the second and third numbers be multiplied together, and the product be divided by the first ; it is evident, that the answer remains the same, whether"^ the number compared with the iirst be in the second or third place. Thus is the proposed demonstration completed. There are four other methods of operation beside the general one given above, any rf which, when applicable, performs the work much more concisely. They are these : • 1. Divide the second term by the first, multiply the quotient by the third, and the product will be the answer. 2. Divide the third term by the first, multiply the quotient by riie second, and the product will be the answer. 3. Divide the first term by the second, divide the third by the quotient, and the last quotient will be the answer. 4. Divide the first term by the third, divide the second by the quotient, and the last quotient will be the answer. The general rule above given is equivalent to those, which are usually given in the direct and inverse rules of three, and which are here subjoined. The Rule of Three direct teacheth, by having three num- bers given, to find a fourth, that shall have the same proportion to the third, as the second has to the first. RULE. I. State the question ; that is, place the numbers so, that th« first and third may be the terms of supposition and demand, and the second of the same kind with the answer required. 2. Bring I08 ARITHMETIC. 3. Multiply the second and third terms together, divide, the product by . the first, and the quotient will be the answer. Note u n 1 M I ■ .^ f 11 — ■ * 2. Bring the first and third numbers into the same denomina* tion, and the second into the lowest name mentioned. ' 3. Multiply the second and third numbers together, and divide ,W the product by the first, and the quotient will be the answer to the question, in the same denomination you left the second numbor in ; which may be brought into any other denomination required* EXAMPLE. If 241b. of raisins cost 6s. 6d. what will 18 frails cost, eacl| weighing net 3qrs. i81b. ? 241b. : 68. (Sd. : : 18 frails, each 3qrs. i8ib. ; 12 28 • 78 "' 102 18 816 102 1836 7B 14688 12852 12) ?4)i432o8 ( 5967 232 ■ 160 2,0)49,7 3 168 Ans. 24I. 17s* S^' £'H 17 3 IThe rule is founded on this obvious principle, that the magni- tude or quantity of any effect varies constantly in proportion to the varying part of the cause : thus, the quantity of goods bought is in proportion to the money laid out ; the space gone over by an uniform motion is in proportion to the time, &c. The truth of the PftOPORTION. 109 "Note i. It is sometimes most convenient to multiply •and divide as in compound multiplication and division | •and sometimes it is expedient to multiply and divide ac- cording to the rules of vulgar or decimal fractions. But when •the role, as applied to ordinary inquiries^ may be made very evi- W dent by attending only to the principles of compound multiplica- tioo and division. It is shewn in multiplication of money^ that the price of oncy multiplied by the quantity, is the price of the whole ; and in division, that the price of the whole, divided by the quantity^ is the price of one. Now, in all cases of valuing goods, &c. where one is the first term of the proportion, it is plain, that the answer, found by this rule, will be the same as that found by multiplication of money ; and, where one is the last term of the proportion, it will be the same as that found by division of money. In like manner, if the first term be any number whatever, it is plain, that the product of the second and third terms will be great- er than the true answer required by as much as tlie price in the second term exceeds the price of one, or as the first term exceeds an unit. Consequently this product divided by the first term will give tlie true answer required, and is the rule. There will sometimes be difficulty in separating the parts of complicated questions, where two or more statings are required, and in preparing the question for stating, or after a proportion is wrought ; but as there cap be no general directions given for the management of these cases, it must be left to the judgment and experience of the learner. The Rule of Three inverse teacheth,by having three num- bers given, to find a fourth, that shall have the same proportion to the second, as the first has to the third. If more reqiure morey or less require /«/, the question belongs to the rule of three direct. But if more require kss, or lesf require more, it belongs to the rule of three inverse* Note. no ARITHMETIC when neither of these modes is adopted, reduce the com- pound termsi each to the lowest deuomihation mentioned in it| and the first and third to the same denomination i then will the answer be of the same denomination with the second term. And the answer may afterward be brought to any denomination required. . Note 2. When there is a remainder after division, re- duce it to the denomination next below the last quotient^ ' and divide by the same divisor, so shall the quotient be so many of the said next denomination ; proceed thus, at long as there is any remainder, till it is reduced to the low* est denomination, and all tiy quotients together will be . the answer. And when the product' of the second and third terms cannot be divided by the first, consider that product as a remainder after division, and proceed tb re*- duce and divide it in the same manner. Note 3. Note. The meaning of these phrases, " if more require more^ kss require Ar/x," &c. is to be understood thus : more requires more^ when the third term is greater than the first, and requires the fourth to be greater than the second ; more requires less, when the third term is greater than the first, and requires the fourth to be kss than the second ; less requires more^ when the third term is lless than the first, and requires the fourth to be greater than the second ; and less requires lessj when the third term is less than the first, and requires the fourth to be less than the second. RULE. 1. State and reduce the terms as in the rule of three direct. 2. Multiply the first and second terms together, and divide their product by the third, and the quotient is the answer to the ques- tion, in the same denomination you left the second number in. The method of proof, whether the proportion be direct or in- verse, is by inverting the question. EXAMPLE. i PROPORTION. Ill Not;;e 3. If the first term and either the second or Aird can be divided by any number, without a remainder, let them be divided, and the quotients used instead of them. Direct znd inverse proportion are properly only parts of the same general rule, and are both included in the pre- ceding. Two or more statings are sometimes necessaYy, \rhich' may always be known from the nature of the question. The method of proof is by inverting the question. ' EXAMPLES. . \r . Et AMPLE. What quantity of shalloon, that is 3 quarters of a yard wide, will line 7i yards of cloth, that is li yard wide ? I yd. 2qrs. : 7yds. 2qrs. :: 3qrs. : 4 4 6 30 6~ 3)180 4)60 i^ yards, the answer. The reason of this rule may be explained from the principles of compound multiplication and division, in the same manner as the direct rule. For example : If 6 men can do a piece of work ia 10 days, ip how many days will 12 men do it ? As 6 men : 10 ikys : : 12 men z ^ » £= 5 4£2jr/> the answer* And here the product of the first and second terms, that is, 6 times 10, or 60, is evidently the time in which one man would perform the work ; therefore 1 2 men will do it in one twelfth part of that time, or 5 days ; and this reasoning is applicable to any Dtber iastance whatever. lia ARJTHMETICf. EXAMPLES. I. Let It be proposed to find the value of 140Z. Sdwt* of goldy at 3L 19s. ltd. an ounce. oz. £ ^. <!. oz. dwt. I : 3 ^9 I' : : 14 8 : 20 20 20 20 79 288 12 959 288 ^672 7672 1918 • 2,0)27619,2 13809x1 pence, or i2)i38o9d. 2TVq- 2,0)115,08. 9d. ^4^' Ans. 57I. I OS. 9d. 2^q- Explanation. The three terms being stated by the general rule, as above, the second term is reducedf ^ to pence, and the fhird to penny-weights, these being their lowest denominations, as directed in the first note. The fitst term h also i^uced to dwts. that it may agrec^ with the third, by the same note. The second term is then multiplied by the third, and the product divided by the first, according to the general rule, when the answei' comes out 13809 pence, and 12 remaining ; which' remainder being reduced to farthings, and these divided b^ the satne divisor 20, by the second note, the quotient is a farthings, 8 remaining. Lastly, the pence are. divided by 12', to reduce them to shillings, and these again by 20 for pounds; when the final suiri comes out 57!. los. 9d. '2q'- for the answen 2. How PHOPORTION. 11 J 2. Hdw much of that in length which is 4^. inches broady #ill make a square foot ? Breadth. Length. Breadth. 4 '5 : 12 : : 12 : 12 in. 4-5) 1 44*0(3 2 = 2f. Sin. the answer. 135 90 3, At^io^-d. per lb. what is the value of a firkin of btfttet containing 561b. ? lb. d. q. . lb. I : 10 2 : : 56 ,- 8 56=^x7 700 7 £2 9 00 the answer; Or thus : % d. d. lb. T • ^^ — T • ' I • y X V =^V^=s88d.=49S.=2l. 9s. as before. Or thus : "ib. d. lb. i : 10 '5 : : $6 '^ 1 0*5 560 i2k88-o a>o)4,9 £2 9 as before, 4. If I 114 ARITHMETie. 4. If -^ of a yard cost -V ^^ ^ pound, what will -^ o€ an English ell cost ? Fh'st f of a yard = f of $ of | = 52i±2ii =:if of an ell. 5X1x5 Then ^ ell : t^l. : : 7^ eU : And ^ X ^ X*^ ^L2iiJ<i-.. .23 =: 9s. 8d. y the answef. 5. If -g- of a yard cost 4- of a pound, what will -^ of a» English ell cost ? i = -375 f = -41. iell = TVyd— "Si^S •37Syd- • '41. ' • '3 1 25yd. : •3^25 *37S)**25oo{*333> &c. = 6s^ 8d. the answer. 1125 1250 1 125 1256 1125 6. What is the value of a cwt. of sugar at ^jA. per lb. ? Ans. 2I. I IS. 4d. 7. What is the value of a chaldron of coals at 1 1^. per bushel ? Ans. il. 14s. 6d, 8. What is the value of a pipe of wine at 10^. per pint ? Ans. 44I. 2s. 9. At 3I. 9s. per cwt. what is the value of a pack of wool weighing 2cwt. 2qrs. 131b. Axis. 9I. 6d. tVt- 10. What is the value of ijcwt. of coffee at 54^1. per ounce ? Ans. 61I. 12s. II. Bought FROPORTION. 115 11. Bought 3 casks of raisins, each weighing 2cwt. 2qr8. 251b. what will they come to at 2L is. 8d. per cwt. ? Ans. i7l.4id.T'rr. 12. What is -the value of 2qrs. inl. of velvet at 19s. S^d. per English ell ? , Ans. 8s. lo-^d. |-^. 13. Bought 12' pockets of hops, each weighing icwt. 3qrs. 171b. ; what do they come to art 4I. is. 4d. per CM-^t. ? Ans. Sol. i2s. li-d. -^V 14. What is the tax upon 7.45I. 14s. 8d. at 3s. 6d. in the pound ? Ans, :i3ol. los. o^. '^~-. 15. If -|^ of a yard of velvet cost 78. 3d. how many yards .can I buy for 13I. 15s. 6d. ? Ans. 28-^ yards. 16. If an ingot of gold weighing plb. 90Z. ladwt. be worth 4^il.^28. what is that per grain ? Ans. i-^d. 17. How many quarters of corn can I buy for 140 dol- lars at 4s. per bushel ? Ans. 26qrs. 2bu. . 1 8. Bought 4 bales of clothj each containing 6 pieces, and each piece 27 yards, svt 16I. 4s. per piece ; what is the value of the whole, and the rate per yard ? Ans. 3 8 81. 1 6s. at 12s. per yard. ^[p. If an ounce of silver be worth 5s. 6d. what is the price of a tankard, that weighs ilb. looz. lodwt. 4gr. ? Ans. 61. 3s. pi-d. ^^. 20. What is the half year's rent of 547 acres of land at 1 58. 6d. per acre ? Ans. 2 III. 19s. 3d. 21. At I '750. per week, how many months board can I have for lool. ? Ans. 47m. 2w^ -j^. 2;i. Bought 1000 Flemish ells of cloth for 90I. how must I sell it per ell in Boston to gain lol. by the whole ? Ans. 3s. 4d. 23. Suppose a gentleman's income is 1750 dollars a year, and he spends 19s. 7d. per day, one day with another, how much will he have saved at the year's end ? Ans. 167I. I2S. id. 24. What Il6 ARITHMETIC. 24. "What is the value of 172 pigs of lead, each vreigli- 5ng 3cwt. 2qr8. l^iib, at 81. 17s. 6d. per fother of 19— cwt. ? . Ans. 286L 4s. 4fd. 25. The rents of a whole parish amount to 1750I. and a rate is granted of 32I. i^s. 6d. what is that in the pound? .' Ans. 4^-d. -^fill^. 26. If keeping for mj horse be ii4<l« per day, what will be the charge of 1 1 horses for the year ? Ans. 192I. 7s. 84-d. 27. A person brea||y|k owes in all 1490L 5s. lod. and has in money, goods and recoverable debts, 784L 17s. 4d. if these tilings be delivered toJiis creditors, what will they get in tlie pound ? " Ans. los. 6^. J^i6io * vhen QC2 28. What must 40s. pay toward a tax, when Q52I. 13s. 4d. is assessed at 83I. 12s. 4d. ? Ans. 5s. i^. -f - ^ I ^ ^ . 29. Bought 3 tuns of oil for 151I. 14s. 85 gallons of which being damaged, I desire to know how I may sell the remainder per gallon sp as neither to g^in nor lose by the bargain? Ans. 4s. 6^d. ' 5 6 11' 30. What quantity of water must I add to a pipe of mountain wine, value 33I. to reduce the first cost to 4s. 6d. per gallon ? Ans. 20j gallons. 31. If 15 ells of stuff -^- yard wide cost 37s. 6A, what 'rt^ill 40 ells of the same stuff cost, being yard wide ? Ans. 61. 13s. 4cL 32. Shipped for Barbadoes 500 pairs of stockings at 3s. 6d. per pair, and 1650 yards of baize at is. 3d. per yard, and have^ received in return 348 gallons of rum at 6s. 8d* per gallon, and 7501b. of indigo at is. 4d. per lb. what re- mains due upon my adventure ? * Ans. 24I:* 12s. 6^. 33. If 100 workmen can finish a piece of work in 12 days, how many are sufficient to do the same in 3 days ? Ans. 400 men. 34. How many yards of matting 2ft. 6in. hroad will cover a floor, that is 27ft. long and 2Dft. broad ? Ans. 72 yards. 35. How MiopoRTioN- try I *35. How many yards of cloth 3qrs. wide arc JH^aLia jneasure to 30 yards 5qr8. wide ? Aiis. 50 yards* 36. A borrowed of his friend B 250I. for 7 months, promising to do him the like kindness : sometime after l^ liad occasion for 3 col. how long may he kccp% to receive full amends for the favour ? Ans. 5 months and 25 -days. 37. If, when the price of a bushel of wheat is 6s«^d. the penny loaf weigh 90Z. what ouRh^t to^eijptjvh^n wheat is at 8s. 2yd. per bushel ? ^^^B VS\ns. 6oz. I3dr. 38. If 4-|-cwt. may be carriSr5^miles for 35s. how ma- ny pounds can I have carric^^o miles for the same money ? ^ Ans. 9071b. -V 39. fjlw^any yards of aanvas, that is ell wide, will line 20 yards of say, that is 3ars.%vide ? Ans. 12yds. 40. If 30 men can perform^ piece of work in 1 1 days, how many men will accomplish arfother piece^of work, 4 times as big, in a fifth part of the timt ? Ans. 600. 41. A wall, that is to be built to the height of 27 feet, was raised 9 feet by 1 2 men in 6 days : how many men must be employed to finisli the wall in 4 days at the same rate of M'crking ? Ans. 36. 42. If -^oz. cost ^-A, what will loz. cost ? Ans. il. 5s. 8d. 43. If -~ of a sliip cost 273I. 2s. 6d. v/hat is -r— of her " worth ? - Ans. 227I. 12s. id. /14. At i^. per cwt. what dces.3Ub. come to ? Ans. io-|-d%, ^t Ans.^4ol. 46. A person, having ^- of n coal mine, sells -^- of his diare for 171I. whiit is the whole mine worth ? Ans. 380L 47. If, when the days are 13-^ hours long, a traveller per- forms his journey in 35— days -, in hew many days will he perform" the sam? journcv, when the d.jys are 1 1--^- hours long ? Ans. 4^^^! •• ^'7^- 4^. A AZ. If f- of a eallon cost -^1. what will ^ of a tun ^t ? I ^:t4^ ARITHMETIC. itM^A r.\:Iaiait of soldiers, consistiag 6f 976 men, are 7^r>c »tv docikcd, each coat to contain 2^ yards of cloth t>vM te I ^-y^ ^*ide, and to be lined with shalloon -^jrd. wide i K^« nuiiy yards of shalloon will line them ? .^ Ans. 4531yds. iqr. ^4^ CTICE. onffifftlOH netnc ' Practice ii5 a coiffiRfSB of the rule of threcj when the first term happetis to be tk unitj or one ; and has its name from its daily use amo^B merchants and tradesmen, being an easy and concise memod of workirij^most quest- ions, that occur in trade andjjbusiness. The method of proof is by the rule of three. An aliquot part of any riffnber is such a part of it, as, being taken a certain-HfUmber of times, exactly makes that number. GENERAL RULE.* I. Suppose the price of the given quantity to be iL is. or id. as is most convenient ; then will the quantity itself be the answer, at the supposed price. 2. Divide * The rule, and its application to the following particular ^ #L«iscs, will be rendered very evident by an explanation of the ex- anipJ^ In this example it is plain, that the quantity 526 is the answ(^at il. consequently, as 3s. 4d. is the ^ of il. i of that quantity, or 87L 13s. 4d. is the price at 3s. 4d. In like manner, as 4d. is I'o of 3s. 4d. so -rV of 87L 13s. 4d. or 81. 15s. 4d. is the answer at 4d. And by reasoning in this way 4L 7s, 8d. will be shewn to be the price at 2d. and los. i i-Jd. the price at ^. — Now as the sum of all these parts is equal to the whole price (3s. lo^d.) so the sum of the answers, belonging to each price, will he thf iii^.swcr at the full price required. And the same will be tru'j ill .i:i; i^amplc v/hatever. i. Divide Ae given price into aliquot parts, either of die supposed price, or of one another, and the sum of the quo- tients, belonging to each, will be the true answer required. Note. When there is ^y fractional part, or inferior denomination of the quantity, take the same part of the price, that the given fraction, or inferior denomination, is of the unit, of which the price is given, and add it to tlie price of the whole number. EXAMPLlfc^ -,. • What is the value of 526 yartWf cloth at 3s. loid. per yard ? 526I. Ans. at iL 3s. 4d. is 7 =z 87 13 4 do. at o 3s. 4d. 4d. is tV = 8 15 4 do. at 4 id. is t = 4 7 8 do. at 2 Jd. is -y = o 10 lit do. at aj 101 7 3i: do. at o 3 io:J: the full pricc^ Ans. I oil. 7s. 3id. A few of the many cases, that may occur, will, with their particular rules, be sufficient to illustrate the general rule. CASE I. When the price is less than a penny, RULE. Divide by the aliquot parts of a penny, and then by la and 20 5 and it will give the answer required. EXAMPLES. I. 4506 at :J. i IS i 2253 ;} is i 1126s i2)3379i :'. . 2,0)28,1 7 £14 I 7i the answer^ 2. 34S(S J 12m ARITHMETIC. 2. 3455 at -J. Aris. 3I. las. 3. S46 at -^. Ans. 2I. I2S. icy^i 4- 347 ^t T- Ans. 14s. 5-i-il; 5. 810 at -?. Ans. 2I. I OS. 7~d. CASE II. Wlen the price is an aliquot part of a shilling. RULE. Divide the given number by the aliquot part, and the quotient is the ans^mL in shillings, which reduce into jp^uiids, as before. ^^ EXAMPLES. 2. 3d. is :j: 172ft at 3d. 2,0)43^2 ;^2i I2S. the answer. a. 437 at id.' Ans. il. i6s. ^i* 3. 5275 at 2d. ♦ Ans. 43I. 19s. 2d. 4. 352 at i-^d. Ans. 2I. 4s.' 5. 1776 at 3d. Ans. 22I. 4s. CASE Jll.- When the price* is a part ^ but not an aliquot part ^ of a shilling, RULE. Divide the given number by some aliquot part of a shill- ing, and then consider what part of the saiii aliquot part the rest is, and divide the^quotient thereby ; and the last quotient, together with the former, will be the answef irtr shillings, which reduce into pounds, as before. EXAMPLES. I. 876 at 8id. 6d. is i 438 3d. is y 146 4 is t 36 2,0)62,0 6 £-^1 o 6 the answer. a. 37i PRACTIC£# tU a. 372 at i|<k Ans. 2I. 148. jd. 3. 2700 at 7-5J. Ans. 8il. us. 3d' 4. 827 at 4fd. Ans. 15I. I OS. i^d. 5. 1720 at ii-jd. Ans. 82I. 8s. 4d. CASE IV. When the prici is any number of shUlings under 20. ' RULE. 1. If^hen the price is an even number^ multiply the given number by \ of it, doubling the first figure, to the right" hand for shillings, and the rest at-e^^unds. 2. When the price is an odd number^ find for the greatest cir€n number, as before, to which add ■-—. of the given num-' bet for the odd shillings^ and the sum i| the answer. EXAMPLES. I. 243 at 4s. 2 348I. 204I. 250I. 48] 2. IS. is ^ 2757 at IS. 872 at 8s. 372 at IIS. 264 at ipSr I. i2s. the answet. 566 at 7s. . 3 169 i6 28 6 .»: A' 6. 198I. 2s, the anstvttr, • - Ans. Ans. . Ans. Ans. 16% 12s. 16s. OTHER I-XAMPLES IN PRAdHCt. 1. 2344 at 5s. Sd, 5s. is I 58 10 6d. is tV S ^7 2d. h \ ^ ^9 2 10 for \ I S for i 70!. I'S. the answer. a. 8cwt. 122 ARITHMETrC. a- 8cwt. Zip9. i61b. at 2l. 5s. , 6d. 8 i8 4 1 2 9 5 8t 9* zqrs. is i 141b. is j; 2lb. is 4 19I. 13s. 3d. the answer. 3- 273^ at 2S. 6d. Ans. 34L 38. xvd* 4. 937T at 3I. 178. 8d. Ans. 3640L 12s. 6i^ 5. 37cwt. 2qrs. 141b. at 7I. ros. pd. per cwt. Ans. 283I. lis. xifd- # TARE AND TRETT- Tare and Trett are practical rules for deducting cer* tain allowances, which are made by merchants and trades- men in selling their goods by weight* ^are is an allowance made to the buyer for the weight of the box, barreli or bag, &c« which contains the goods bought, and is either at so much per box, Sec at so much per cwt. dr at so much in the gross weight. T^rttt is an allowance of 41b. in every L04lb. for waste, dust, &'c. Cloff is an allowance of 2IK upon every 3cwt. * . Gross wight is the whole weight of any sort of goods, together with the box, barrel, or bag, &c. that contains them. ZuttU is the weight, when part of the allowance is de- ducted from the gross. Net luelght is what remains after all allowances are made. CASB * TARE AND TRETT. 1 23 CASE I. if^hen the tare is a certain iveight per hoxy barrel^ or bag^ i^c. RULE.* Multiply the number of boxes, or barrels, &c. by the tore, and subtract the product from the gross, ani the re- mainder is the net weighi required. ;' ' )[. In 7 fraih of raisins, each weighing 5cwt. 2qrs. 51b. jgross, tare 231b. per frail,^ how much net ? >;}X7s?icwt. i^r. ailb. -i . ' ' ' : CWt. 4i#.9i. ' • 3^ 3 7 'grtsfeV" ■■ I I 21 tare. 37 I 14 ^ic»6wer. 2. In 2/Vi barrels of figs, each 3qrs. jplb. gross, tare jplb. per barrel, how many pounds net ? Ar^s. 22413. • ^ What is the i^et weight of k 4. hogsheads of tobacco, fadi 5cwt 2C[rs. 171b. gross, tare xoolb. per hhd. ? Ans. 66cwt. 2qrs. i^lb. : CASE 11. ff^hen the tare is a certain weight per cu't. f RULE. Divide the gross weight by the aliquot parts of a rwr. contained in the tare, and subtract the qxnotient from the gross, and the remainder is the net weight. EXAMFLES. * It is manifest, that this, as well as every other case in this rale, is only an application of the rules of proportion and practice? Zi4 ARITHMETIC. EXAMPLES. 1. Gross I73cwt. 3qrs. 171b. tare i61b. per cwt. how much net ? cwt qrs. lb. ' «73 3 ^7 pos$, f nil y4lb. is i ai 2 2$ • jilb. U 4 3 o II 24 3 9 . 44^ o 8 tbeanswen «. Wh$t is the net weighjof 7 barrels, of .pot^ash^. each ureighing 20 lib. gross^ tacie fle^ng at lolb. per cMrt. ? Ans. 1 28 lib. 60Z, 3. In 25 barrels of figs, each 2cwt. iqr. gross, tare i61b, per cwt. how muclf af^ i . Ans. 48cwt. 241b, "'case III,.. ^ fP7>en Trett^is allowed with Tars. ■■ RULE, f - J .. i.' Divide the suttle weightby 2*6, and the quotient is the trett, which subtract from the -suttle, and the remainder i^ the net weight. \^ - - . • ■ EXAMPLES. I., In pcwt. 2qrs. i7lly. gross," tare 371b. and trett as wsual, how much net? cwt. qrs. lb. 9 ^ ?7 gfoss. 019 tare. 26)9 o 8 suttle. ' I II trett. 8 2 25 the answer. 2« In 7 casks of prunes, each weighing 3cwt. iqr. 5lbj gros3, tare i7Tlb. peV cwt. and trett as usual,, how much net ? Ans. igcwt. 2qrs. 25!!:^, 3. What COMPOUND PROPORTION. I2j; * - • 3. What is the net weight of 3 hogsheads of sugar weighing as follQWS : the first, 4cwt. 51b. gross, tare 731b. ; the second, 3cwt. 2qrs. gross, tare 561b. and the third> ^cwt. 3qrs. 171b. gross, tare 471b. and allowing trett to each as usual ? Ans. 8cwt. 2qrs. 4lb, CASE. IV, When tare^ trett and doff arn all allowed* RULE. Deduct the tare and trett, as before, and divide the sut- tk by i6§, ^nd the quotient is the cloflF, which subtract from the suttle, and the remainder is the net. EXAMPLES. I. What is the net weight of a hhd. of tobacco, wcigh- Jng 15cwt.3qrs.20lb.' gross, tare 71b. per cwt. and trett ^nd cloff as usual ? cwt. qrs. lb. ' 15 3 20 gross. ^: 7I9. JS tV 3 ^7 tare. 26)14 3 21 2 8 trett. .168)14 I 13 suttle. 9 cioir. 14 I 4 the answer, 2. Ir\ 19 chests pf sugar, each containing i3cwt. iqr. 171b. gross, tare 131b. per cwt. and trett and cloff as usual, how rnuch net, and what is the value at 5-|:d. per pound ? Ans. 2i5cwt. 171b. and value 577I 6s. r^d. COMPOUND PROPORTION. Compound Proportion teaches how to resolve such questions, as require two or more statings j Simple rro- portion. — ^ ia6 ARITHMETIC, In ihcse questions there is always givcfi an odd number of terms, as five, seven, or ni::e, &c. These arc distin- guished into terms of supposit'tQti^ and terms of demand^ the number of the former always ea^ceeding that of the lat- ter by one, which ]^ of the ss^me Jcind with th> term or answer sought. This rule is often named the Double Rule of Threcy be- cause its questions are sometimes petforn^ed by tvro oper^ tions of the rule of three, RUL£^ FOJt STATING. 1. Write- the term of supposition, which is of the same kind with theanswef, for the middle term. 2. Take one of the otbepr terms of supposition, and one pf the d/emanding terms of the same kind ^with it \ then place one of th^m for a first term> and *tHe. other for t third, according to the directions given in the rule of three. Do the san^e with another terni of supposition and l^ts correspondent demanding term \ and so on, if there be more terms of each kind % writing the terms under each other, which fall on the same side of the middle term. METHOD OF OPERATION. I. By several operations, ^-^'Hzkt the two Upper terms and the middle term, in the same order as they stand, for the first stating of the rule of three ; then take the fourth number, resulting from the first stating, for the middle term, and the two next terms in the general stating, in the same order as they stand, for the extreme terms of the second * The reason of this rule for stating, and of the methods of operation, may be easily shewn from the nature of simple propor* tion ; for every line in this case is a particular stating in that rule. And, therefore, with respect to the second method, it is evident^ tliat, if all the separate dividends be collected into one dividend, and all the di'-* ors into one divisor, their quotient must be the answer sought:- COMPOUND PROPORTION* iij iecond stating ; and so on, as far as there are ainy numbers in the general stating, always making the fourth number, resulting from each simple stating, the second term of the next. So shall the last resuhing number be the answer required. 2. By one ^^rj/w«.— -Multiply together all the terms in the first place, and also all the terms in the third place. Then multiply the latter product by the middle term, and divide the result by the former product ; and tlie quotient will be the answer required. , Note l. It is generally best to work by the latter meth- od, viz. by one operation. And after the stating, and be- / fore the commencement of the operation, if pne of the first terms, and either the middle term, or one of the last terms, can be exactly divided by one and the same number, .«^, let them be divided, and the quotients used instead of " them 5 which will much shorten the work. ■ ^ Note 2. The first and third terms of each line, if o£.i difierent denoniinations, must be reduced to the same de- nomination. EXAMPLES. 1. How niany men can complete a trench of 135 yards Jon^ in 8 days, provided 16 men can dig 54 yards in 6 days ? General Stating. 54 yds. or 2I . g . . ("135 yds. or 5"! . 8 days, or4i * ^^^^^ " I 6 days, or 3 J ' First Method. yds. men. yds. day*, men. days. >*Sf y4-5-27=:2 2 ; 16 :: 5 : 4 : 40 : : 3 : i^ 135-5-27=5 5 3 «^ 2=4 i. 6-f- 2=3 2)80(40 m^n, 4)120(30 meo, the answer. 8 X2 .. . R Second Method. :} ■■-■■■■ U] 4 » : i6 i6 8)240(30 nien> ike. answer as before, 24 o , 2. If lool. In one year gain 5I. interest, what will be the interest of 750L for 7 years ? Ans. 262I. los* 3. What principal will gain 262L los. in 7 years, at 51*- per cent, per annum ? Ans. 75 oL l^ 4. If a footman travel 130 miles in 3 days,- when the days iare 12 hours long 5 in how many days, of 10 hour« ■*^'Cach, may he travel 360 miles ? Ans. 9|-|-days. "^ 5". If 120 bushels of corn can serve 14 horses 5<5days i^ how many days wil> 94 bushels fecrve ■6' horses ? Ans. ro2~ day^. 6. If 70Z. 5dwts. of bread be bought at 4-|d. when corn 18 at 4s. 2d. per bushel, what weight of it may be bought for IS. 2d. when the price of the bushel is JS. 6d. ? Ans. lib. 40Z, 3 J-rytU*'tni 7. If the carriage of I3cwt. iqr. for 72 miles be 2I. los. 6d. what will be the carriage of 7cwt. 3qrs..for 112 miles ? Ans. 2I. i%. I id. i-rrg^* 8- A wall, to be built to the height of 27 feet, was^ vs raised to the height of 9 feet by 12 men in 6 days j how . many men must be employed to finish the wall in 4 days, at the same rate of working ? Ans. 36 rAeu. 9. If a regiment of soldiers, consisting of 939 men, can cat up 35 r quarters of wheat in; 7 months ; how jmany soldiers will eat up 1464 quarters in ^ months, atTlivjirrate ? CONJOINED ?UOPOaTlON. 1291 10. If 248 men, in 5 days of 11 hours each, dig a trench 230 yards long, 3 wide and 2 deep ; in how many days of 9 hours long, will 24 men dig a trench of 420 yards long, 5 wide and 3 deep ? Ans. 28 8 v™. CONJOINED PROPORTION. Conjoined Proportion is when the coins, weights, or measures, of several countries are compared in the same question ; or it is the joining together of several ratios, and the inferring of the ratio of the first antecedent and the last consequent from the ratios of the several antece- dents and their respective consequents- . Note i. The solution of questions, under this rille, may frequently be much shortened by cancelling equal numbers, when in both the columns, or in the first column and third term, and abbrevfaiij^g those that are commens- flrable. Note 2. The proof is by so many statements in the single rule of tJire^Tas the nature of the qucbtion requires, CASE I. When it is required to find how many of the last hind of coin^ weight, or measure, mentioned in the question, are equal to a given number of the first, i RULE. I. Multiply continually together the antecedents for the first terni, and the consequents for the second, and muke the give^number the third. a. ThW find the fourth term, or proportional, which will be the answer required. luSAMPtES. R ^ ■"^' • t30 ARITHMETIC. EXAHiPLES. I. If lolb. at Boston make plb. at Amsterdam ^ polb. at Amsterdam, ir2lb. at Thoulouse ; how many pounds at Thoulouse are equal to 501b. at Boston i Anu Cods. 10 : 9 90 : 112 900 : ioo8 : : 50 : 59 )504oo(56 the answer. 4500 5400 54PO Or by abbreviation. ^ 10 : 9 :: 50 10 : i :: 50 i : i ti g 90:112 xo:xx2* 10:112 2 : 112 :: r j6 56 the answer. 2. If 20 braces at Leghorn be equal to 10 vares at Lis^ bon ; 40 vares at Lisbon to 80 braces at Lucca ; how ma« ziy braces at Lucca are equal to joo braces at Leghorn ? Ans. 100 braces*^ CASE n. When it is required to find how many of the first kind of cwt^ weight, or measure, mentioned in the question, are equal to a given number of the last. ^RULE. Proceed as in the first case, only make the produ^rt of the consequents the first term, and that of «the antecedents, the second. ^ . EXAMPLES. r * In perfonniDg this example, the first abbreviation is obtained by dividing 90 and 9 by their commoD measure 9 ; the second by dj^vidiog 10 and 50 by their common measure 10 ; the third by dividing 10 and 5 by their common measure 5 ; and the fourth, or answer, by dividing 2 and i f 2 by their common measure 2* •^ BAaTEHJ ^ 131 £XiMPL£S. I. If loolb. in America make pjlb. Flemish f and xplb. FlemUbj 251b. at Bologni^ 5 how many pounds in Ameri- ca are ec^ual to 501b. at Bolognia ? Cons. Ant. 95 : 100 ^S : 19 A1S 190 «37f ; 1900 : : 50 f 50 )95ooo(4olb. the answer. 9500 , o Or by abbreviation. J>5 I 100 5 : 100 5:4 iS% 1^ if 5^ ^5* i::5o i;i;:50 1:4:: 10; 4 Ans. 4oU). a. If 251b. at Boston be 22lb. at Nuremburg ; 881b. at Nuremburg, palj). at Hamburg ; 461b. at Hamburg, 491b. at Lyons ; how many pounds at Boston are equal to p8Ib. at Lyons ? Ans. loclb. 3. If 6 braces at Leghorn make 3 ells English ; 5 ells English, 9 braces at Venice ; how many braces at Leghorn ^l make 45 braces at Venice ? Ans. 50 bracet. BARTER. Barter is the exchanging of one commodity for anoth- er, and directs traders so to proportion their goods, that neither party may sustain loss. RULE. X32 AUITHMETIC. RULE.* FfndtKe value of that commodity, whose quantity is giv- en ; then find what quantity of the other, at thcrrate pro* posed, you may have for the saint moneyj and it gives the answer required. EXAMPLES.* 1. How many dozen of qandl^s at js. 2d. per, dozen, must be given in barter for 3cwt. 2<jr8. of tallow at 37s* 4c!. per cwt. f J qr. 8. 4 '• 37 12 d. 4* cwt. qr. • - : : 3 ^ 1' * ■ 4 448 14 1792 448 M \ 4)6272 •12)1568 ..... • . • ^ ■ ■ : : .. 2,0)13,0 8 61. s. d. s 2 -. (52 I OS. doz. I 8d." ■ 1. s. d, : ; 6 10 8 ( 20 •■ X30 12 62)1568(25 124 '" 328 3» 12 loz. and 3. 62)216(3 . - . 186 30 2 2. How"" *. This rule is evidently, only, an application of the rule of three. LOSS AN'D«GAIN. 1 33 a. How much tea at 98. pef lb: can I have in barter, for 4 cwt 2^s. oi chocolate at 4s. pw lb. ? Ans. 2cwt, 3. How many reams of paper at 28. p^-ci. per ream must be given, in barter, for 37 pieces of Irish cloth at iL 12s. j\d, per piece ? Ans. 428|4* 4. A delivered 3 hogsfie:\ds of brandy at 6s. 8d. per gal. to B, for J 26 yards of doth ; what was the cloth per yard ? ■ Ans. I OS. 5. A and B barter ; A hath 4icwt. of hops at 30s. per cwt. for which B gives him 20I. in money, and the rest in prunes at 5d. per pound ; what quantity of prunes must (A receive ? Ans. I7cwt. 3qrs. 41b. 6. A has a quantity of pepper, weight net i6oolb. at I7d. per lb. which he barters with B for two sorts of goods, the one at 5d. the other at 8d. per lb. and to have — in moneyjp and of each sort of-goods an equal quantity : how inany pounds of each must he receive, and how much in Jfwntj ? Ans*^i394|4lb. and 37I. i5«. 6jd. LOSS AND -GAIN, Loss AND Gain is a rule that discovers what is got or lost in the buying or selling of goods ; and instructs mer- chants and traders to raise or lower the price of their goods, §a as to gain or lose a certain sum per ctnt. &c. ■* Questions in tliis rule are performed by the Rule of Three. EXAMPLES. • S34 AftXTaMBTIC SXAMt»tiES. I. How must I sell tea per pound, that cost me 139. 5^^ to gain at the rate of 25 per cent. ? 1. 1. s. d. 100 : 125 :: 13 5 2 12 805 3« 161 1,00)201,25 12)201 25 1 58. pd. -^ the answer. Or thus ; 4)135. 5d. ' • 3 4 1 6s. ^fd. the same as before. 2. At 3s. 6d. profit in the pound, how much per cent. ? Ans. 17L IDS. • 3. Bought goods at 4-|k1. per lb. and sold them at the rate of 2I. 75. 4d. per cwt. what was the gain per cent. ? Ans. X2l. 13s. I id. 4. Bought cloth at 78. 6d. per yard, which not proving so good as I expected, I am resolved to lo^ 1 7-^ per cent, by it : how must I sell it per yard ? Ans. 6s. 2-^. 5. If I buy 28 pieces of stuffs at 4I. per piece, and sell 13 of the pieces at 61. and 8 at 5I. per piece ; at what sate per piece must I sell the rest to gain 20 per cent, by the whole ? Ans. 2I. 6s. xojd. 6. Bought 40 gallons of brandy at 3 s. per gallon, but by accident 6 gallons of it are lost ; at what rate must I sell the remainder per gallon, and gain upon the whole prime cost at the rate of 10 per cent. ? Ans. 38. lofd. 7. Sold )FSLLOWSHZF. 13^ 7. Sold a repeating watch for 175 dollars^ and by so do- ing lost 17 per cent, whereas I ought in dealing to have cleared 20 per cent, how much was it sold for under the just value? Ans; 23L 8s. o-^d. FELLOWSHIP. Fellowship is a general rule, by which merchants, &c. trading in company, with a joint stock, determine each per- son's particular share of the. gain or loss in proportion to his share in the joint stock. By this rule a bankrupt's estate may be divided among his creditors i as also legacies adjusted, when there is a de- ficiency of assets or eflPects. SINGLE FELLOWSHIP. Single Fellowship is when difierent stocks are employed for any certain equal time. RULE.* As the whole stock is to the whole gain or loss, so 19 each man's particular stock to his particular share of the gsun or loss. METHOD OF PROOF. Add all the shares together, and the sum will be equal to the gain or Joss, when the question is right. EXAMPLES. * That the gain or loss, in this rule, is in proportion to their stocks is evident : for, as the tunes the stocks are in trade are equal, if I put 10 \ of the whole stocky I ought to have f of ^e whole gain 5 if my part of the whole stock be f , my share of the whole g;dn or loss ought to be §- also. And, generally, if I put in i of the stock, I ought to have ^ part of the whole gain or loss ; that is, the same ntto, that the whole stock has to the whole gain pr loss, must each person's particular stock have to his particular ^n or loss. 13(9 AUTHTMETICa EXAMPtES* ,./•' I. Two persons trade together 5' A put into $tock isdJ, and B 220L and they gained 500L what is each person^s share thereof ? . 130+220=350 350 : 500 : : 130 : 130 . l/ooq 500 35,0)6500,0(185!. " 300 = . 200, . 25 20 J5)500(i4». 150 10 .. ra 35)i2o(3d. IS 4 35)<So(iq. 2S •350 : 500 : : 220 ; 220 r 1 0000 1000 35,0)11000,0(3141. 50 150 JO 20 35)200(55. 25 185I. 14s. 3-id. 4^=:A's share. 12 . 314I. 5s. 8id. |^=B's share. 35)3oo(8d. *50oK os. od. the Proof- 20 . ■■.'.. ■- J 4 • » , 35)8o(2q. 10 FEUiOWSHIp. 137 2. A and B have gained by trading 182I. A put into stock 300I. and B 400I. what is each person's share of the profit ? Ans. A 7 81. and B ip4l, 3. Divide 120L between three persons, so that their shares shall be to each other as i> 2 and 3, respectively. Ans. 2oL 40L and 6ol. 4. Three persons make a joint stf>ck ; A put in 1841* los. B 961. 15s. and C 76I. 5s. they trade and gain aaoL I2S. what is each person's share of the gain ? Ans. A 1 13I. i6s. 4t4» ^ 59^^ ^4S. ttti C 47I is. -^; 5. Three merchants A, B and C, freight a ship with 340 tuns of wine ; A loaded no tuns, B 97, and C the rest. In a storni the seamen were obliged to throw 85 tung overboard ; how much must e.ach sustain of thfi^ loss ?' A 274-, S 24^, And: €33^. 6. A ship worth 860I. being entirely lost, of .which — belonged to A, ^ to B, and the rest to C ; what loss wiU each sustain, supposing 500L of her to be insured ? Ans. A 45I. B 90I. and C 225I. 7. A bankrupt is indebted i:o A 275I. 14s- to B 304I. 7s. toC 152I. and to D 104I. 6s. His estate is worth only 675I. 15s. how must it be divided ?. Ans. A 222I. 15s. 2d. B 245I. i8s. i-j-d. C 122I. 163. 2-|d. and D 84I. 5s. 5d. 8. A and B, venturing equal sums of money, clear by joint trade 154I. By agreement A was to have 8 per cent, because he spent his time in the execution of the project, and B was to have only 5 per cent. ; .what was A allowed for his trouble ? Ans. 35I. los. 9-i^d. DOUBLE FELLOWSHIP. Jlouhle Fellowship is when different or equal stocks ar^ employed for different times. \ S.VX.5* S I3t^ AftlTHMETtC* RULE.* Multiply each man's stock into the time of its continue ance, tKen say. As the totaJ stm of i|ll the products is to the whole gain or loss. So is each man's particular product to his particular share of the gaiu or loss. EXAMPLl^S. r. A and B hold a piece of ground in oommcm, for which they are to pay 361. A put in 23 oxen for 27 days, and B 21 oxen for 35 da,ys v what part of the rent ouglit each man to pay ? . . 23x27=621 1356 : 36 : ; 62* : 21X35 =735. - 621 ^356 . 1356)22356(1(8-. 8?3<5f 660 20 f 356)1 32oo(9S. 12204 996 12 *356)"953^(W- 10848 1 104 1356)4416(35. 4068 348 i35<S * Mr. MALCOLir» Mr. Ward^ and several other authors, have {^veo an analytical iovcstigation of this rule : but the most general and nu^wsHir^ zjp 1356 -: 36 :': 735 73J 180 .108 252 ■ 1356)215460(191. 1356 ... 12900 . 12204 » ' 696 20 ^. .k . . . . ' 356) 1 3920(1 OS. # 360 12 16I. 9S. 8|<l.TMr5=A's share. 19I. los. 3d. 4^f|=B's share. i3S^)432o(3<l- 4068 36I. OS. od. the Proof, 252 4 ' •' ■ ^ - J 1008 " 2. Three graziers hired a piece of land for 6ol. los. A put Jn 5 sheep for 4^: months, B put in 8 for 5 months^ and C put in 9 for 6j- months : how much must each pay of the rent ? * Abs. A iiL 5S.-B 20I. and C 29I. 5s. . 3. Two merchants .enter into partnership for 18 months ; A put into stock at first 200I. and at the end of 8 months hie put ifl lool. more ; B put in at first 550!. and at the end ef 4 months took out 14QI. Now at the expiration of tlicr '*'^ time a^d elegant method, perhaps, is that, vi^hich Dr. Hutton has giv- tB in bis Arithiaetic, viz. . . When the times are equal, the shares of the gain or loss sR-e evi- dcptly as the stocks, as in Singje Fellowship '; and w^en the ftocks are. equal the shares are as the ^mes ; wherefore, \|rhen nei* llier are equal, the shares must be as their products. * /. 142 * ARITHMETIC. 5. A refiner melts lolb. of gold of 20 carats fine with i6ib. of 18 carats fine ; how much alloy must he put to It to make it 22 carats fine ? Am, It is not fine enough by 3-^ carats, so that no al- loy >must be put to it, but more gold. ALLIGATION yiLTERNATE. jliligrJion altervnie is the method of finding what quan- tity of any number of simples, whose rates are given, will compose a mixture of a given rate ; so that it is the re-- verse of alligation medial, and may be proved by it. RULE I.* . T. V/rite the rates of the simples in a column under each other. > ■> 2. Connect * Demonstration- By connecting the less rate .to the greater, and placing the differences between them and the mean rate alternately, the quantities resulting arc such, that there is pre- cisely as much gained by one quantity as is lost by the other, and therefore the gain and loss upon the wHole are equal, and* are ex- actly the proposed rate : and the same will be true of any other iwo simples, managed according to the rule. In like manner, let the number of siAples be what it may, aii4 witli how many soever each is linked, since itis-^ways a lefiis.with a greater than the mean price, there will be an equal balance of loss and gain between every two, and consequently an equal bal- ance on the whole. Q^ E. D. It is obvious from, the rule, thajt qwestknis of this sort admit ,of a great variety of answers ; for, having found one answer, -we . may find as many more as we please, by only multiplyiijg or di- viding each of tlie quantities found by 2, 3, or 4, &c. the reasofi of which is evident ; for, if two quantities of two simples make a balance of loss and gain, with respect to the mean price, so must also . \ \ iLLIGATION ALTERNATE. M3 a. Connect or link with a continued line the rate of each simple^ which is less than that of the compound, with one or any number of those, that are greater than the com- pound ; and each greater rate with one or any number of the less. 3. Write the difference between the mixture rate and that of each of the simples opposite the rates, with which they are respectively linked. ^ 4. Then if only one difference stand against any rate^-it will be the quantity belonging to that rate ; but if ^hero be several, their sum will be the quantity. EXAMPLES. I. A merchant would mix wines at 14s. 19s. 15s. and 22S. per gallon, so that the mixture may be worth i8s. the gallon : what quantity of each must be taken ? ' V8 4 at 14s, 1 at 15s. 3 at 19s. jf at 22s; Or thus : 1+4 at 14s. at 15s. 4+5 ? *^ '9s- 4 4 at 22s. 2. How much wine at 6s. per gallon and at 4s. per gal- Ion, must be mixed together, that the ^composition may be worth ss. per gallon ? Ans. 12 gallonsji or equal quantities of each. 3. How; also the double or treble, the 4 or §■ part, or any other ratia of these quantities, and so on, ad tnfiniium. Questions of this kind are called by algebraists indeterminate or unUrmted problems, and, by an analytical process, theorems voxf be raised, that will give all the pombk answers. 144 ARITHMETIC. 3. How much corn at 2s. 6d. 3$. 8d« 4$. and 48. 8d. per bushel^ must be mixed together^ that the compound may be worth 3s. lod. per bushel ? Ans. 12 at '28. 6d. 12 at 3s. 8d. 18 at 4s. and 18 at 4s* 8d. 4. A goldsmith has gold of 17, 18, 22 and 24 carats fine : how^ much must he take of-each to make it 21 carats fine ? Ans. 3 of "ly, i of i8| 3 of 22 atid 4 of 24, 5. It is required to mix brandy at 8s. wine at 7s. cider at IS. and* water 2tt o per gallon together, so that the mixt* ure may be worth 5s. per gallon ? Ans. 9 of brandy, 9 of wine, 5 of cider, and 5 of water., mULE 2.* Wi>en tie whole composition is limited to a certain quantity^ find an answer as before by linking ; then say, as the sum of * A great number of questions mt^X be here jgiyen relating to the specific gravity of metals, &c. but qo^ of the most curious, with the operation at large, may serve as a^su^icient specimen^ Hi£RO, king of Syracuse, gave orders for a crown to be made entirely of pure gold ; but suspecting the workmen had debased it by mixing it with silver or copper, he recommended the discove- ry of the fraud to the famous Archimedes ; and desired to know the exact quantity of alloy in the crowi^ Archimedes, in order to detect the imposition, procured two other masses, one of pure gold, the other of silver or copper, and each of the same weight with the former ; and each being put separately into a vessel full of water, the quantity of water expell- ed by them determined their specific bulks : from which and their given weights, the exact quantities of gold and alloy in the crown may be determined. Suppose the weight 6f each crown to be lolb. and that the wa- ter expelled by the copper or silver was •921b. by the gold 'jzlb. and by the compound crown *64lb. what Will be the quantities of gold and alloy in &e crown* f The ALLIGATION ALTEKKATE. I,;. of the quantities, or differences thus determined, is tirthc given quantity, so is each ingredient, found by linking, to the required quantity of each. EI^MPLES* I. How many gallons of water at os. per gallon must be mixed with wine worth 33. per gallon, so as to fill a vessel of 100 gallons, a^d that at gallon may be afforded at ^s. 6d. ? 6 3^5 : 100 : : 6 : 36 • 100 : : 30 S6 : 100 : : 6 : 36 : 100 : 6 30 36)600(16 36 36)3000(83 288 240 216 X20 108 24 12 Ans. 83y|- gallons of wine, and i6j^ of water. 2. A grocer has currants at 4d. 6d. pd. and i id. per lb. and he would make a iftixture of 2401b. so that it may be afforded The rates of the simples are 92 and 52> aod of the compound 64 ; therefore Cgz — I 12 of copper, 1^2 — " 28 of gold. And the sum of these is 1 2 -|- 2 8=40, which should have bcca but 10 ; wheoce, by the rule, 40 : 10 :: 12 : 31b. of copper 1 ,, — ,— lo : 10 : : 28 : ?lb. of goS? j *« '^^''• T ^-'Affi AIUTHMETIC. aircr^ed at 8d. per pound i how much of each ' sort must he take ? Ans. 72lb. at 4d. 24 at 6d. 48 at pd. and 96 at i Ld« 3. Ho^ much gold of 15, of 17, of 18 and of 22 ca- rats fine, must be mixed together to form a composition of 40 ounces of 20 carats fine ? Ansj. 50Z. of 15, of 17 and of i8j and 25 of 22. RULE 3.* " W/jen one of the ingredients is litmted to a certain quantity ,* take the diiFerence between each price and. the mean rate as before ; then, As the difference of that simple, whose quantity is given> is to the rest of the differences severally, so> is the quanti?. ty given to the several quantities requiredi. EXAMPLES. I. How much wine at 5s. at 5s. 6d. and 6s.. the gallon, must be mixed with 3 gallons at 4s. per gallon, so that the: mixture may be worth 5s. 4d. per gallon ? '48- 8+2=10 • r48 .r — ^1 I 66 1 L72— 8 + 2=10 i6+4=2a 16+4=20 10 10 10 10 20 20 3. 6 6 Ans. 3 gallons at 5s. 6 at 5s. 6d. and 6 at 6s. 2. A ♦ In the.very same manner questions may be wrought, when BCveral of the ingredients are limited to certain quantities, by find- ing first for one limit and then for another. The two last rules can want no demonstration, as they evident- ly result from the first, the reason of which has been already explained. ^ INVOLUTION. 147. ^ A grocer would mix teas at 12s. los. and 6s. with iolb. at 4s. per pound ; how much of each sort must he take to make the composition worth 8s. per lb ? Ans. aolb. at 4s. 10 at 6s. 10 at los. and ^o at I2S. .3. How much gold of 15, of 17 and of 22 carats fine, must be mixed with 50Z. of 18 carats fine, so that the composition naay be 20 carats fine ? Ans. 50Z. of IS carats fine, 5 of 17, and 25 of 22« JNVOLUTION. A fov^'K is a number produced by multiplying any giv- fia number continually by itself a certain number of times. / Any number is itsdf called the /rst power ; if it be jDiultiplied by itself, the product is called the second power, or the square y if this be multiplied by the first power again, the product Is called the third power, or the cube ; and if this be multiplied by the first power again, the product is called the fourth power, or biquadrate ; and so on ) that 2S, the power is denominated from the number, which ex- ceeds the inultiplipations by i. Thus, 3 is the first power of 3. 3X31:? 9 is the second power of 3. 3X 3X3:1:27 is the third power of 3. 3X3X3X3=81 is the fourth power of 3. &c. &c. And in ^this ipasner is calculated the following table of powers. TABLE Ufl AUTHMCTIC. TABLE 9fAi first twlve PoWEits tff the 9 DiGJTB, 00 • 1^ i 5 00 c< CO Ov 00 CO 1 VO to OS i % CO VO CO 10 M '^ M 00 VO eo VO CN I-- 00 VO V? 1 CO 4: VO c< V6 CO 00 CO Sn CO ON o\ 00 00 •rr % CO 1 ON CO 1 CO 3. OS 00 52 VO CO o\ -\6 C\ M r^ 00 CO * ►^• CO CO M 00 t^ VO >0 % ft NO M SO CO VO ON 1^ 8 1- VO VO us 5? M CO %o M 00 CO M CO 10 N . M 00 N 00 "5- ^ t VO 00 CO NO M 00 CO wo VO 9< VO »o 00 3" CO OS VO VO »-« t CO to ON s- ^ CO 1 to N 'l- 00 VO CO •1- 00 ^ ■ «o 10 8 1 2. - « 1 - 1 M - M - 1 ■-^ •w wm - i Ph •5 Si 2 r4 1 to i •s 1 00 i 2>, Note r. INVOLUTION, 149 Note r. The number, which exceeds the multiplicsif* tions bjT I, is called the index ^ or exponent^ oi the power i «o the indeii: of the first power is i, that of the second power is 2, and that of the third is 3* &c. Note 2. Powers arc commonly denoted by writing their indices above the first power : so the second power pf 3 may be denoted thus 3*, tixe third power thud jSth* fourth power thus 3*, &c. and tlie sixth power of 503 thus 503 ^ Involution is the finding of powers \ to do which wc have evidently the following RULE. Multiply the given number, or first power, contlnuallf by itself, till the number of multiplications be i less than the index of the pbwer to be* found, and the last product firill be the power required,* Note. Whence, because fractions arc multiplied bf taking the products of their numerators and of their de- ^minators, they will be involved by raising each of their |erms to the power required. And if a mixed number be proposed^ * Note. The raising of powers will be sometimes shortened by working according to this observation, viz. whatever two or more powers are multiplied together, their product is the power, whose index is the sum of the indices of the factors ; or if 2t power be multiplied by itself, the product will be the power, whose index is double of that, which is multiplied : so if I would find the sixth power, I might multiply the given number twice by it- lelf Cw the third power, then the third power into itself would Me the sixth power ; or if I would find the seventh power, I might^.first find the third and fourth, and their product would be the seventii ; or lastly, if I would find the eighth power, I might ^rst find the second, then the second into itself would be tho Ibortb, and this into itself would be the eighth. 15« ARITHMETIC. proposed} cither reduce it to an improper fraction, or re» dace the vulgar fraction to a decimal, and proceed by the rule* EXAMPLES* X. What is the second power of 45 ? Ans. 2025, 2. What is the square of '027 ? Ans, '000729, 3. What is the third power of 3*5 ? Ans. 42*875. • 4. What is the fifth power of .029 ?• Ans. '000000020511149; 5. What is the sixth power of 5*03 ? Ans. ^6196^005304479729, 6. What is the second power of j ? Ans. -|> EVOLUTION. The Root of any given number, or power, is such a number as, being multiplied by itself z certain number of times, will produce the power ; and it is denominated the Jlrst^ second^ thirds fourth^ {5V; root^ respectively, as the number of multiplications made of it to produce the given power is o, i, 2, 3, &c. that is, the name of the root is taken from thfe number, which exceeds the multiplication^ \^j. I, like the name of the power in involution. Note i. The index of the root, like that of the power in involution, is i more than the number of multiplications! necessary to produce the power or given number. Note 2. Roots are sometimes denoted by writing ^ before the power, with the index of the root against it : 3 80 the third root of 50 is ^Z 50, and the second root of it is ^ 50, the index 2- being omitted, which index is always understood, when a root is named or written without one. But if the power be expressed by several numbers with the sign -|* or — -, &c. between them, then a line is dratirn from EVOLUTION. 151 from tjie top of the sign of the root, or radkal agn, over. all the parts of it : so the third root of 47—15 is 3 ; -— . ^47 — 15. And sometimes roots are designed Kkc po^vcrs, with the reciprocal of the index of the root above JL the given number. So the 2d root of 3 is 3 * ; the 2d root of 50 _L » * is 50^ •, andthe tlurd root of it is 503 ; also the third root of \ 47 — 15 is 47-— I5p. And this method of notation has justly prevailed in the modem algebra ; because such roots, being considered as fractional powers, need no other di- 'rections for any operations to be made with them, than those for integral powers. Note 3. A number is called a complete power of any kind, when its root of the same kind can be accurately ex- tracted ; but if not, the number is called an imperfect pow- er, and its root a surd or irrational number : so 4 is a coi»- plete power of the second kind, its root being 2 ; but an imperfect power of the third kind, its root beiijg a surd number. Evolution is the finding of the roots of numbers either accurately, or in decimals, to any proposed extent. The power is first to be prepared for extraction, or evo- lution, by dividing it from the place of units, to the left in integers, and to the right in decimal fractions, in- to periods containing each as many places of figures, as arc denominated by the •index of the root, if the power con- tain a complete number of such periods : if it do not, the defect will be either on the right, or left, pr totli ; ^^ the defect be oh the right, it may ]>c supplied by a ri Hexing cyphers, and after this, whole periods of cyphers "^ay be annexed to continue the extraction, if necessa- ry 5 but if there be a defect on the left, such defective period ^ust remain unaltered, and is accounted the first period of ^^-e given number, just the same as if it were complete. Now t5i ARrrHMErxc. Now thi« division may be conveniently made by writing m point over the place of units, and also over the last fig- ure of every period on both sides of it j that is, over every second figure, if it be the second root ) over every third, Jf it be the third foot, &c* Thus, to point this numbet ito^$Bg6' 12^25 > for the second root, it will be 21035896' 127350 } but for the third root 2io35896*i2735o ; • .... auid for the fourth 2io35896" 12735000. Note. The root will contain just as many places or figures, as there are periods or points in the given power ; and tliey will be integers, or decimals respectively, a8#thc periods are so, from which they are found, or to which they correspond ; that is, there will be as many integral or deci- mal figures in the root, as there are periods of integers or decimals in the given number. To EXTRACT THE SqUARE RoOT. RULE.* X. Having distinguished the given number into periods, find a square number by the table or trial, either equal to, or < .1 .. I , . I . , ■ ■ . .. .11 . ■ ■ ■ ■ I .. 1 1— . — * 111 order to shew the reason of the rule, it will be proper to premise the following LcMMA. The product of any two numbers can have at most but as many places of figures, as are in both the factors, and it least but one less. Demonstration. Take two numbers consisting of any num- ber of places, but let them be the least possible of those places, viz. unity with cyphers, as 1000 and 100 ; then their product will be I with as many cyi)her5 annexed as are in both the numbers, viz. **^ EVOLUTION. XJ3 or the next less than, the first period, and put the root of it to the right hand of the given number, after the manner of a quotient figure in division, and it will be the first fig- ure of the root required. ^. Subtract the assumed square from the first period, and to the remainder bring down the next period for a dividend. 3. Place viz. lOQOOp ; but 1 00000 has one. place less than 1000 and 100 together have : and since 1000 and 100 were taken the least pos- sible, the product of any other two numbers, of the same number of places, will be greater than looooo ; consequently the product of any two numbers can have, at least, but one place less than both the fiictors* Again, take two numbers of any nuaj}er of places, that shall be the greatest of those places possible, as 999 and 99. Now 999X99 is less than 999 X 100 ; b»it 999 X 100 (=99900) con- tains only as many places of figures, as arc in 999 and 99 ; there- fore 999 X 99 or the pfoduct of any other two numbers, consisting of the same number of places, cannot have more places of figures than arc in both its factors. Corollary i. A square number cannot have more places of figures than double the places of the root, and, at least, but one ' less. Cor. 2. A cube number cannot have more places of figures than triple the places of the root, and, at least, but two less. The truth of the rule may be shewn algebraically thus : Let A^=: the number, whose square root is to be found. Now, it appears from the lemma, that there Will be always as many places of figures in Ac root, as there arc points or periods in the given number, and therefore the figures of those places may be represented by letters. Suppose 1^4 ARITHMEnC. , 3. Place the double of the root, already founds on the left hand of the dividend for a divisor. 4. Consider what figure must be annexed to the divisor, so that if :be result be multiplied by it, the product may be equal to, or the* next less than, the dividend, and it will be the second figure of the root. j. Subtract the sa'd product from the dividend, and to the rcm;iincltr Itfing down tlie next period for a new divi. dend. d. {"ind Supposes N to cousist of two periods, and let the figures ia tht root be represented by a arid-^. Then a+h =:/!* + itfJ+^*=3^= gi^en number ; and to find the root of N is the same as finding the root of a^4-2«^4«3^, tkc method of doing which is ae fisUows : 1st dm^ov a)a^+2ab+b^{a+h=:rcfqU ^ 2d divisor 2a'^-b)iah+by Again, suppose N to consist of 3 periods, and let the figures of the root be represented by «, b aftd c. Then fl+^+r r:^* + 2^3+3 *4-2jf+2^f+c% and the miR« ner of finding //, b and r, will be zt before : thus, » ist divisor a)a* + 2^-f-3* +2<3f-f-2fe-f-c*(a4-3+t=: root. 2d divisor 2a-^b)2ab^b^ 2ab+b* ' 3d divisor 2a-\-2b'\'c)2aC'\-2be'^c^ 2ac+2bc+c^ Now, the operation, m each of these cashes, exac^y agrees with the rule, and the same will be foupd to be true when A'' consists of any number of periods whatever. ^EVOLUTION. 155 6. Find 3 divisor as before, by doubling the figures al- ready in the root 5 and from these find the next figure of the rootj as i^ the last article ; and so on through all the periods to the last. Note I. When the roc>t is to be extracted to a p;reat number of places, the work may bfe much abbreviated thus : having proceeded in the extraction by the common method tm ybu have found one more than half the- required num- ber of figures in the root, the rest may be found by divide ing the last remainder by its corresponding divisor, annc^p- ing a c)''phcr to every dividual, as m division of decimals ; or rather, withput annexing cyphers, by omitting cpntvpu- ally the right hand figure of the divisor, after the manner of contraction in division of (tgciitiAls^ Note 2. By means of the square root wc readily fini the fourth root, or the eighth root, or iht sj^cteenth root, &c. that is, the root of any power, whose index is some power of the nun^ber 2 ; namely, by e::tr icfing so often the square root, as is denoted by that power of 2 ; that is, twice for the fourth root, thrice £ox the eigUtli root, and so on. To EXTRACT THE SQUARE R60T OF A VUL- GAR Fraction. ' tvi^i^ •-•■ First prepare all vulgar fractions by reducing them ta their least terms, both for this and all otimr roots. Then 1. Take the root of the numerator ami that of the denominator for the respective terms of the root required. And this is the best way, if the denoxninatof be a complete power. But if not, th^ 2. Multiply the numerator and denominator together ; take the root of the product : this root, being made the numerator to the denominator of the given fraction, or the denominator ^ ♦ 156 AWTHMETIC. denominator to the numerator of it, will form the frao^ tional root required, fO^ ^ah a And this rule, will serve, whether the root be finiti or in- finite. That is. v/i==^'=: Or 3. Reduce the vulgar fraction to a decim^j and ex-* tract its root» EXAMPLES. ^ T. Required the square root of 5499025, . : • • • • . 54990*5(2345 *« root. 4 43I149 3|J29 464I2090 411856 4685 23425 23425 2. Required the square root of 184*2, • • • • i84-2ooo(i3*57 the root, I 23| m 265I1520 •51^3^5 2707 19500 18949 "^ 551 remainder. 3. Required 3- Required the squa/e root of 2 to 12 places^ " 2(1-41421356237 + root. • 24! 1 60 41 96 a8i I 400 281 2824111900 4111295 28282160400 2I56564 2S2841 1383660 11282841 • , ^ » I I i» m 9828423110075900 3I 8485269 2828426 ) 159063 I ( 56237 + ' 176418 169706 - at" . ' '■ 6712 1055 849 i^ " ' ■ 206 198 8 H^ What is the square root of 15239902; f Alls. 12345- ^« What is the square root of '00032754 i Ans. 'oiQ^. 6. What 159 ARITBMfiTiC; 6. Whit ia the square root of ^^^ ? Am. •54S497' 7. Wbit is the square root of (^ ? An?v 2-5298, &c, S. What is the square root of xo ? ff Ans. 3' 162277, &c, # To EXTRACT THE ClTBE RoOT* RULE.* u Having divided the given number intb periods of j figures, find the nearest less cube to the first period by the table of powers or trial ; ^et its root in the /}uotient, and Subtract the said cube from the first period 5 to the re- mainder bring down the second period, and call this the resolvend. '. .' 2. To three times the square of the root, just foandt add three times the root itself^ setting this one place more to the right than the former, 904 call this sum the divisor. Then divide the resolvend, wanting the last figure, by the divisor, for the next figure of the root, which annex to the former ; calling this last ^gure e, and the part of the root before found call a, 3« Ada * The reason of minting the given number, as directed in the rule, is obvious fronv Cor. 3, to the' Lemma made use of in de* monstcating l^ie square root ; and the rest of the operation will be best understood from the following andytical process : Suppose i\r, the given number,; to consist of two periods, and let aic figures IB the root be denoted by a and b, Thco 3. Add together these three products, namely, thrice the square of a multiplied by ii thrice a multiplied by the square of e^ and the cube of/, setting each of them one place more • to the righc hand than the former, and call the sum the subtrahend ; which must not exceed the resolvend ^ wul if it do, then make the last figure i less^ and tt^ut the operation for finding the subtrahend. '^ anJFtc T 4* From the 'resolvend take the subtrahend, anJPto thc' remainder join the next period of the given number for a new resolvend ; to which form a new divisor from .the whole root now found ; and thence another figure of the root, as before, &c. EXAMPLES. *rheh j+i|*r:fl'+3tf*34-3fl^*+3'=:Afe given numlxr, and to find the cube root of A^ is the same as to find die cube root of «' 4-3^1 '3 -1*3^^^+^' ; the method of doing which is as follows; ^'+3^*3+30^* +i* ( tf+terroot^ 3^2*^4- 3if3*+^' resolvend. 3«* 3«* +3^ 'divisor. 3^^^+3rz3*+*' subtrahend. And in die same manner may the root of a quantity, cooslr.iDg gf any number of periods whatever, be found. 1^9 ' AUTHICETIC. £XiMPLES. - X. *To extract the cube root of 48228*5441 .3X3* = 27 3X^3 = .09 Divisor 279 48228'544(36*4 root. 27 ?i : — X3*X6 = 162 1 X3 X6*= .324 f 65= 216 > 2T228 rcsolvcnd. add 3X36* =38.88 3X36 =t: 108 38988 19656 subtraliend* 1572544 resolvend. 3X36»X4 = 15552 3X36 X4*= 1728, ^ add 4' = 643 1572544 subtrahend. 2. Whit id the cube root of 1092727 ? Ans. 103. ' 3. What is the cube root of 27054036008 ? Ans, 3002. 4. What is the cube root of '0001357 ? Ans. •05138, &C, 5. What is the cube root of -jffl'? Ans. f .. 6. What is the cube root of j ? Ans. '873, &c/ ! Rule for extracting the Cube Root by I. Find by trial a cube, near to the given number, and ^ call it the supposed .cube. £ 2. 'Thenl * That this rule converges extremely fast may be jcasily shew; tbus : ■■ . a. Then, twice tte'awj^jpdtcd cube iddcd to the given number, is to twice the g^ven number added to the sup« iposcd cube, as the root of the supposed cube is to the root required nearly. Qr as the first sum is to the di£Ference of the given and supposed cube, so is the' supposed root to the difference of the roots nfcarly. 3. By taking the cube of the root thus found p)T the supposed cube, and repeating the operation, the t^ will b« had to a still greater degree of exactness. i. If is required to find th^ cube root of 980^449^ Let Let N=t given number, a' s^ supposed cabe, and xzs oor- lectiouU ^ I Then 2a'+iV : zN+q^ ti a ; a^-x.by the rufc, and coo« seqnemly xa^+Nxa^^sztN+a* x^i (x 2a^ +a+x^ x a+0 sziN+a^Xa. Or za^+za^x+a^+^^x + &i*«» + 4^' + x^Sz2aN+a\ and by transposing the terms, and dividing by za Nzza^ +^a*x+3ax* +x^ +x^ -i , which by negkct^ig th« zn t X* terms x' + — , as being very small, becomes Nssa^ ^.j^^x.^ za Sax*+x*ss the known ctfbe of m+x. Q. E. L Let i3}oM<MidSLwnp<Mllfii)^;V^ Then x25obcoad ' 98003449^ "' ' ^"' - " •- *' "i 250000000 * 195006898 98003449 125000000 T- 348ot>3449' : 321606898 r f goa : • ■ ■ ; ' ■ 1 - " - t^reoC'iMrff.' 348003449) 160503449000(461 = corrected root, i3»2P7379(^,: - 2x3020694^ 20)58020694 421862460 , «- . . , .24l8Q02iLi£lL • 'i . .'--•■■ -. Js'SjfpoxJ - • ^ -v . .■ .= «.• Required the cnbc tootW 2i035'8,r Here we so^'SyfiSthat tlicroot Hcs between 20 and '30^ its cube is 19683 the assumed cube. Then , __ 19683 2I035'8 39366 4207.1*6 21035-8 19683 As 6o4oa*8 : 6x754.6 : : 27 : 27*6047 -> -,•: ^T --J ^7 - . ■ ".: 4322822 . „ :■_... -. v: -:-• 1235092 4i — — 16673 74*f 6030 x*8) ''•• ' 1208036^, . , .., ... . / • *:* 4228 r J iu . 3^525 36241 ■ t ' j;^2 42 -^ . Again for a second operatioiTi tfui c^ 6f this root U 41035-3 18645 1 J;5JBg3, »4 t^pJPCpw by tthp latter meth- pd is thus : ..'.._ 4207o*6j729Q' ^f^rS . 21035-8 21035-318645, &c. As (J3lotf*437^$ '^ d**K -491355 2: 27'<J»47:thPdMr.2ai r / *QO«2 10834 ' . ' . t)ie root require^ 3, What is the. cube root of 157464 ? Ans. 54. 4. ^What is the cube root of |- f Ans. -'763, &c. 5- What is *e cute fddt of iiy ? * ■ Ans. 4'89t97. To EXTRACT THE RoOTS OF PoWEtS IN cenerAl. RULE.* , I. Prepare thd* given number for extraction, by poindng off from the units place as die root required-directs. 2. Find * ThisLrule wiD ba sufficieQ^y obri<^ bom, the work in the ibUowing es^ample ; ^ Entrac- i |tt power zroni tbe givw nuinbfv* 3. Tq the Tdpiaiader bring dowff Ae -first figure in tliQ 4. Involre ZmMH the cube root of a^+6£' — 40«' 4-9$9— 64. '^ * " ' tf*-|-&i'-*40«' 4-960-^(41* ^-itf— 4 ' . '■ . . , ' ' . -. ~:i ? # • -••-'• *" , J t ■ . H^en the index of the power^ ivhofie ro^ is to be tx^tWJei^ l§ a'compdsite noliiber, the fblSovitig role wiQ be tenriceable : '' Take any two or more hidicesy whose product is the given in- dent and extract oat of the given nomber a root answering to one ^f ttiete iodiccs } aUd then oat of this loot extract a root answer^ bg to sndcher of the indicesi and so 00 to the' last. Thus, the foorth root 3? sqaare root of the square root^ ' vThe sixth root =: square root of the tx/^ ro9t» 8c^ The proof of! all roots is by involution. The following theorems may sometiines be found useful in eacr tracting ^ root of a vulgar fraction iV^-TS^£:l^=r — f-| mnivcrsally, tfl* _^jL _ elTx a . I* [ . 4* lAfoIvt dii-root to the next inferior power to lhat| which is given, and multiply it by the namber denptifig ihc givdrf'powcr for a /fiv//^. 5. Find hpw many tim^ Ae divisor may be had in the dividend, and the quotient will be another %ure of the yoot. ' 6* IfwolTe the whole root to the given p6weri a»d tubr tract it from the given number as before. 7/ Bring down the first figure of the next period to th^ repiainder for a z|ew dividen^i to MT^hjch find a new diytp^t^ jind so on, till the whole be Shished. I. yn^\ is tie <?ubc root of SSt^yjyfi.f • • • 5315737^376 ^7=3^ 3*X3?=27)2t^i dividend. — n*f3 j*X3=4^o7)^S043 second dlvidcn4f S3^S737<> ?. What is the biguadrate root of 1998717337* ? Ans. 37(!i 3. Extract the sursolid; or fifth root, of 307682821 106 715625. .Ans. 3145. ^. Extract . . ■ Am. a753j|, * . - tS. Iipd a^mtoA loot flf »«077*74^3<»7f l?3^^49 Ans. 3aoi7, . . , . ., '^^.:'3S27» To EXTJIACT AHt RodT WH^TBVpk J»T ^PPROJtiMAtlOlff 1. Assume ^ root nearly^ and jraise it to the sfune power with the gireri number^ wKi^h call die assumed power. .... a. Then, as the sum of the aasaBMd power multiplied by the index more i and the given i^yunber multiplied by the index less i, is to the sum of the given number multi* plied by the index more i and the assumed power muL> |:iplied by the index le^ i| so i$ ^ assiiqied^root to the required root. Or, as half the first s^m is to the difference between the given and assumed powers, so if -the aiiisum^ root to the difference between the true and assumed roots ; which difierence, added .or subtracted, gives the true root nearly* And the (Operation maybe repeated as'often as we please, by using always the last found root for the assumed roott ai^d its power as aforesaid for the assumed power* EXAMPLES. I. Required the fifth root of aiQ35'8. Her^ kimM^^' . ityj ♦ Utrc it app^iiii that this fifth rooifc it hdi^mum. 7'3 and t*4* 7'3 hcing taken, its fifth pcwer i$ 20730*7 1593.- Hence then 21035-8 == gtteh humbef . ^ 2073 0*7 1 6 =r assumed powerv " 305'oft4 =; difference. • 5 =5f in4ex. 20730-71^ , 2ilc>35*d 5 + i2=J6 3 2 ' 0^2=r3 ($2i9a^i4S^ 42071*6 4r^2qS2 ^ . 42671*6 104263748=:^ the first sum^ 1042(^3*7 : 36^-6«4 : : 7*3 : •0213605. 73 t • " ' I ■ ■■ ;" ■!■ 915252 2135588 104263*7)2227*1 i32(*o:|i 36043;: di&roiot. • 208527 7*3 ; 141 84 7*321360 = root, 10426 true to the hst Bgutt^ 3758 6^0- 62rf 2. What 19 the thfard root of 2.? . Al» f2599ai. 3. "What is the sixth root of 21035*8 ? Ant. 5'a5403> ' 4> What is the seventh root of 2i03jrS ? Axis* 4'»4S39** J. What 5. Whatittbd aimh root of* ^1035*8? . r. . - • 1 Ana* 3*or22239;» AftlTttMETfCAL PROCRE^SI&I^. Ant rank of numbers increasing by ^ commpn. excejf^, or dteci^asmg by a common differem^e, are said to be in Aritbmttieal. Progression ; such are the numbers i, z» j* Aw 5, &c. 7,'. 5f 3, I ; stod •«> ''6** '4, -2^ When thcmmiberg increase they form an ascending series / but when they de- crease, they form a descending series* The numbers, which form the series, are called the terms of the jirogression/ Any three of the jivi following tcnns being giireif, the Other twcy may be ifeadily founds ' I- The first term, ? commonly called the 2. .The last tenn, . : j extremes. 3. The number of terms. 4* The coftimbh* difference.' S« The suio of all the terms. . Tie first term^ tie last term, and the numbfr of terms iemg ffven, to find the sum of' off the terms. RULE** Multiply the sum of the extremes by the number of t€h»>.«Ad halTthe product will be the answer. EXAMPLES. * Suppose another series of the same kind with the giren pne he plactd onder it in an inverse order ; then will the sum of every two corresponding terms be the same 9s that of the first and last ; consequendy. ARITHMfiTICAL PROCRESSION. l6p EXAMPLES. ** 1. The first tenn of an arithmetical progression is a» the last term 53^ and the number of terms 18 1 required the sum of the series. • J 440 55 2)990 495 _ O^' 53+2X18 * r^^ =495 "*^ answer. 2. The first term is i, the last term 21, and the number of terms 11^ required the sum of the series. Ans. 1 21. 3. How many strokes do the clocks of Venice, which go to 24 o'clock, strike in the compass of a day ? Ans. 300^' 4. If eonsequently, any one of those sains, multiplied by the number of terms, must give the whole sum of the two series, and half thil sum will evidently be the sum of the given series : thus. Let I, 2, 3, 4, 5, 6, 7, be the given series ; and 7, 6, 5, 4,. 3, 2, i, the same inverted ; then 8+8+8+8+8+8+8=:8X7=56and 1+3+4+5 +6+7=1-^=28. Q. E. D. 170 ARITHMETIC 4* If 100 Stones be placed in a right luie, exactly a yard fisunder, and the first a yard from a baskot, what length of ground yrill that man go, who ^th^rs theni up singly, re- taming with them one by one te the basket ? ' Ans. 5 miles and 1300 yards. PROBLEM II. 27v first iertTiy the last ierniy and ih& tiuffiher of terms being give.'iy to find the common' difference, RULE.* DIxdde the difFerencc of the extremes by the number of terms less i, and the quotient will Be the common differ- ence sought. EXAMPLES. I. The extremes are 2 and 53, and the number of terms^ is i3 J required the common difference. 53 ^ 18 , 2 r 17)51(3 n 5^ Or, ^ — = i— =^ the answer. j8— I 17 ^ 2. If * The diiference of the first and I^st terms evidently shews the increase of the first term by all the subsequent additions, till it becomes equal to the last ; and* as the number of those additions is evidently one less than the number of terms, and the increase by every addition equal, it is plain, that the total increase, divided by the number of additions, must give the difference at every one 5«parately ; whence the rule is manifest. L AKITHMETICAL FROC^KESSIOK. 171 2. If the extremes be 3 and -ip, and the number of terms 9^ it is required to find the common difference, and ^he sum of the whdle edries. Ans. The diflference is 2,. and the sum is 99. 3. A mjm is to travel from London-- 1^ a certain place in 1 2 days, and to go but 3 miles the iirst day, increasing every day by an equal excess, M) that tKe last day's journey ^ay be 58 miles ; required the daily increase, and the dis- tance of the place from London. Ans. Daily increase 5,- distance 366 miles. J?ROBJLEM lit. ■ Given the Jirst ierm^ the last term^ and the common djffeitncey to Jind the number of terms, ^.JPivide tli<? difference of the extremes by the common difference, and the quotient, increased by i, is the num- ber of terms required. J ' - ■ ^ EXAMPLES. ^ By tlie last problem ,' the di^ercncc of die extremes, divided by the number of terms less i, gives the comruoa difFcrence .;. consequently the same, divided by the common difference, mast give the number' of terms less i ; hence this quotient, augmented by I, must be the answer to the question. In any arithmetical progressioxu the sum of any two of its terms is equal tp the sum of any other two terms, taken at an equal distance on contraiy sides of the former ; or tlxe double of any one term is equal to the sura of any two termsl taken at an equal distance from it on each side. The sum of any number of terms {n) of the arithmetical series of odd numbers f, 3, 5, 7, 9, 6cc. is equal to the square («*) of that number. That J 7* AKITHMETIC. EXAMPLES, I. The extremes are 2 and 53, and die commbn difier*: ence 3 j what is the number of tenns i ■ S3 2 3)5 1 18 Or, |i~ ..4.i— 18 the answer. 2. Jf That isy if It 3y 5> 7, 9^ &c. be the numbers^ Then will i, 2^> $% 4*9 5^9 6cc. be the soms of i^ z, $, ^c^ of those terms* Tot, 0+ 1 or the sum of x term == i ^ or i X-j-3 or the som of 2 t«rms =: 2* or 4 4+5 or the sum of 3 terms = 3* or 9 9-I-7 or the sum of 4 terms =: 4^ or i6, 8cc* Whence it is plain, that, let n be any number ^atsoever, the •am of n terms will be n^. The following table contains a sun^^iary of the whole doctrinq of arithmetical progression. c ASES OF ARJTHMEriCAL PROGRESSION. \ Case Giv. jRcq. Solution. X. adn < / I»— IX^+O. t iiX^i+iJ^X-. 2 Case ABJTHMETJCAL P&OG&ESSION. 173 2. If the extremes be 3 and ipi and the comaion diflfer- cnce 2p vhat is the number of terms ? Ans. 5^ 3- A Case z. 3- Giv* |R.^. Solution. n ft -+.. zd _ - - ^ za — d\ + ^d^ — 2a — d i 5/ 3tf— 4 +8*6 — J a 4. cis s 4 ^ 2S , 5- 6. am < (iitt < d I 2j a. d i ^+/x« 2 . ^ - ■ — -^ — a — - Case 174 i^THMETIC. 3. A man, going a journey, travelled the first day 5 miles, the last day 35 miles, and increased his journey ^cry day by 3 miles j how many days did he trarel ? ^ Ans. II days. GEOMETRICAL fc4se ^m 1 Re- 5: Solution. 7- a Jnl . — l~^—i K ^ s 8. snd * a I a n 3 n Z 10. dh ^ 2 n d id n 2X«/— J n — r X« Here< a = least term. n = number of terms. s zz sum of all the terms. d = common difference. / = greatest term. - — ^ ■ ■ . ... J GEOlCBTKICiL PR0GR£S6I0N. 1 75 GEOMETRICAL PROGRESSION. Aky series of numbers, the terms of which gradually increase or decrease by a constant multiplication or divis- ion, is said to be in Geometrical Progression, Thus, 4, 8, 16, 32, 64, &c. and 81, 27, 9, 3, I, &c. are series in geo- metrical progression, the one increasing by a constant mul- tiplication by 2, and the other decreasing by a constant di- vision by 3. The numbef, by which the series ia^^ constantly increased or diminished^ is called the ratio* PROBLEM I. Given the first terniy the last term^ and the ratio, to find tht sum of the series. RULE.* Multiply the last term by the ratio, and from the prod- uct subtract the first term, and the remainder, divided by the ratio less i, will give the suin of the series. EXAMPLES. ^ Demonstration. Take any series whatever, as i, 3, 9^, a7, 81, 243, &c. multiply this by the ratio, and it will produce Uie series 3, 9, 27, 81, 243, 729, &c. Now, let the sum of the proposed series be what it will, it is plain, that the sum of the sec- <ind series will be as many times the former sum, as is expressed by the ratio ; subtract the first series from the second, and it will give 729 — I ; which is evidently as many times the sum of the first series, as is expressed by the ratio less i ; consequently / ^T" ^s sum of the proposed series, and is the rule ; or 729 is the last term multiplied by the ratio, i is the first term, and 3 — i is the ratio less one ; and the same will hold let the series be what it will. Q^ E. D. NOTIU 17(J MITHMETIC. EXAMPLES^ 1. TKc first term of a series in geometrical progression is I, the last term is ZiBy, and the ratio 3 ; what is ths warn of tlie series ? 2187 3 6561 I 3- -1=2)6560 3280 Or, 3X 3187 — 1__ 3—1 . 3280 the answer. 2. The Note i. Since, in any geometrical series or progression, when it consists of four terms, the jjroduct of the extremes is equal to the product of the means ; and when it consists of three, the product of the extremes is equal to the square of the mean ; it follows, that in any geometrical series, when it consists of an even number of terms, the product of the extremes is equal to the product of any two means, equally distant from the extremes ; and, when the number of terms is odd, the product of the ex- tremes is equal to the square of the mean or middle term, or to the product of any two terms equally distant from them. Note 2. Then 5 c a b h c : d directly, h I d\rj alternation. d I c h^ inversion. C'\'d : J by composition. c — d : d by division. : r : f +^/ by conversion. ^+// mixedly. For in each of these proportions the product of the extremes h equal to that of the means. iSBOiaTMCit FEpCftfiSSXON* X77 a« 'K^^ egkP^^, <^ a |^ine|ncal progres$ion are i and Ans. 87381. 3« Tlie extremes of a geometrical series are 1024 and 59049^ and the ratio is i|- ; what is the sitm of the series ? Ans. 175099. ^ PROBLEM n. "Given the frst tertn and the retk^ ia j/M emj §An^ Usm assigned. R0LE.* i 1. Write down a few of the leading terms pf the series^ «nd pkce 4ieir indiees over dMmi begiaidng with 4 typ^ier. 2. "Add together the most cdnvenient in^icei ^ to make an { Ipdex less bj t than the number exprfsdlng the phce of^ I the term sought. 1 • ' ' • 3. Multiply the tenns of the geome^cal leries ^get|^ ^ff§i Wonging to tfacne indices^ mnd wa^km t$e product 2, Jdiridend. i * BsMOMiTRATiOM. In example z, ^here ^t first teraib ibagnk-tot th e miu» -tibe reason of the n^-is eridenit ; ibr as ereary mi is so^e power ff i^t rat^ and the isdic^ Deinf out t^ie ftomber of Actors, it is p]^n from the n^ofd of iplti|licati^9 j iMrilie prodttct of any two terms wM-be ano^er tcm^ conps* r. 'pondiog with ihe inde^^^ ?^hich is the sum ff ;^e pdices stanc^ng Oier those respectiyc terms. And in the ^secoti4 V^x^nple^ where the series does pot b^a I i»Mi the lado, It appe^«> that ^very term'frftfr the two ^st con- ' ^Mvefore the rale» in this case» is equsJly evident. r/^ The T 178 AVTRinmc* 4* Raue ibf fiat texm to a powcTi wlioie mdeac Is i lesi tlian die namber of terms mvltipliecli and make the result ;. Divide The ibUowiiig tabk concuis sll the ponibfe cases of geometric* ilprogmsioa* Cases of GeOMETHICAL PRQGItESSION, Case Giv. 1 Req. Solution* I. am * rn^K '^-'x-. 3. 5- m4 * 1+^-^ . 1 Ly l—L, a AfJT 4 r n i'rr— [Xf+tf ^1^ i:,r 4-' ad * r s — a LJ^L,a ^, i Case CB01i£T&lCAL PROGRESSION. »79 5* IKvide tbe drridend by the diyisor> ^d the qootieiit unll be die term'sought* Note. When the first tenn of the senes is tqttal ts the ratio, the indices must begin with an unit, and the in* 4iGPt added wut ipakc the entire index of the term re> quired ; 'KcgT Case a?. am ^ J: SoktioD. rs «.— / ml ms /x;=7[""'=«x7r;i*^. L J I J JL i . " ■ ■ J ■ ■ B m ■ ■ ' B * W I J ' lllH n w d * ■■' '*!=? r- /+ r — I X*- r«— 1 x/. Case i8o AmirHUTxc. quired | and the prpduct of tbe dii^nf: t0i% ,filftAd «3 l>efoi^ wi|l giye the tenn xequired. •''-.-. ' ■ *• ... t. IV fii^t tenn of a geometncal aeries Is Of the niunbej: pf tgrms i^y ^ {he ratfo 2 ^ reqdred the last term.. . " « . . f • i> 2> 3, 4» jb, 'indices ; 2> 4i 89 itf> ;32> leading temitiu Xhcn 4-^-4^3-^2 zz index to the k3th ffexm. . And idX 1^X8X4=8192 the^ns^^- ' In this example the indices must begin #i& i^ ^nd isucb of &em be chosen, as will makeoip the ejitire index tk the term required. a* Requiifed Case 9- Gix. IJLegk r/r 20. fJl Solutiofi..- W i>.i ■ » t '^:^x;^ : -I- ■ I ii. ay.s-ar'^lyC!^lY Here f ^ == least term. / = greatest terni. s s£ isutnt)! ail the tenw|* n z= number of terms. r =r ratio. ^ L = Logarithm. CEOMETRfdlL HtOGSCSSION. iS^X 2. &e^t(uitBtl the iltb tMn of a geometriod «erie8> vfUfOt itst term is 3^ trndisMo^. 0> i» 2f 3» 4* 5* *^ ittfliccs. 39 69 I2» 24» 489 96, 1929 leading tenns. Ito 6«|-j Jr index to the 12th tafii. " And I92><96=i8432 = dividend. The number of terms multiplied is 2, and 2 — izzi b the power to ygiach the tfna .3 is to be raised ; but the ^st power of 3 is 3, and therefore i8432-i^3=6i44 the }2Xb, term feq^ifare^ 3. The first term of a geometrical series is i^ the ratio ^ and the numbeir pf imtfii^ 123 } reqmred the la»t term. ' . ' Ans. 4194304* ' ^ . ^irXstiONS TO BE SOLVED BY TH1& TWO PRSCEDIMO P&OBIJIMS. ^i.j. tA person, being asked *to dispose of a fine horse, said be would sell him on condition of having one farthing for die .first nail in his shoes, 2 farthings for the second, one penny for the third, and. so cm, doubling the price of every iei^ tD^32,1the niuhber of nails in his four shoes : .what lirould the horse be sold for iktthBt rate ? Ans. 4473924I. 5s. 3|d. 2. A young man, skilled in numbers, agreed with a farmer to work for him eleven years without any other re- jrard than the produce of one wheat corn for the first year, and, that produce to be sowed the second year, and so on ^m year to year till the end of the time, allowing the in- jcrease to be in a tenfold proportion : what quantity of is^eat is 4ne for sudi^flffeei and to what does it amount^ *^z dollar per bu Ad ? Ans. 2260561* bushels, allowing 7680 wheat corns ^ {» a pint 5 and the amoijnt is 226056^ dpUars.. 3- What . > Vthm Mritr will lie 4ki^^»iq^mJ!^ mmdM^ by VKjmg iL tlie fintjoaaiatl^i. q£. iKc second,'^ 4]^ ^ diirdt and so on^each sacceeding payment being douBte the la»t fc and irlutiHll tike iasfiA^^ Ans. The ddit if 4t^^^m\!btm fmmt ao48l. SIMPLE INTEREST. « • - ^ . . . . IntkIiest is the premius^>albwed-^ Ar.loaii^ money* - The ittin lent is called the principaL The ium ti the principal and interest is called the amount* Interest is allowed at so much per cent, per oimum^ vAaxh premium per cent, per annumy or interest of looL for a year/b called the rate of tnttsMl^ 7 ^ Interest is of two sorts, smple and compound. Simpk interest is ihatj ^Mdk i» aUowed oelf ^ £» dio principal lent. Note. Commissiony Brokerage, Insoranccy. Stocks,* andy in genera], whatever is at a certain rat<^ or sum per ' cent, are calculated Hke Simple Interest. KULE,t I. Multiply the principal by the rat^ and divide the product by 100 ; and the. quotient is the answer for one year. ' 2. Multiply * Sioci is a general name for poblic funds, and capitals of trad- ing companies, the shares of whicb are transferable from one per* son to another. t The role is evidently an appKcadon of Simple Propordoo and Practice. % 0* Middplf tibe bterat far one year by tihe gi^ and the product is the answer for Aat time. 3. U tlicre be parts of a year^ as months or days, work for the months by the aliquot parts of a year, and for the days by Simple noporiion* kXAlIPLBS. 1. What is die interest of 4Sol. for a year> at 5 per cent, per anmim,? 450L S i,oo)aa'5o ao io*oo Ans. aaL and-^=:^=f5aafiosJ 2. What is the interest of 720I. for 3 years, at.5 per cent per annum ? , * 720!. 36. ^ 5 3 36*00 108L AnsJ 3. What IS the interest of 170L for i^ year, at 5 per cent, per annum f 170L 2)8h 108. interest for i ^ear. 5 .45 8-50 12I. 158. Answer^ 26 iO'OO 4. What 4. Wliat IS the Interest of 107L for 117 days, affe 4|^per cent, per annum I 4i . II 7 428 55 18 1 5'« J5 II ^ fl*4 XXX104.7SII7 ^ 559 I 6 o • 3<55)594 13 o ft-4(«t tM-^YtM- ' 1-65 365 the answer. 12 — V — — 229 7'8o 20 4 )4593 q- 4- „ a-4=2f=fa. t 3-20 jPs 5. What 18 the interest of 32I. 5s. 8d. for 7 years, at 4-^ per ceitt {ler annnm i Ans. ^L/ias. i-pL^d^^ 6. What is the interest of 319I. 6d. for 5-^ years, at 3*1 per cjent. per ammm 2 Ans. 681. 154. pd* a^^q. 7. What is the interest of 6p7'5oD. for 5 years, at 6 per cent, per annum ? Ans. i82*25D* 8. What is the interest of 213I. from Feb. 12, to June" 5, 1796, it being leap year, at 2t P^^ c^^^* P®*' aanum ? Ans. 2I. 6s. 6d. s /gVs ^* SIMPLE SIMPLE INT£&£a.T MT WCZMAL8« »8s>, . « SIMPLE INTEREST BY DECIMALS, f RULE.* Mdltiply continually the principal^ ratio and timei and it will give the interest required. Ratio is the simple interest of il. for i yeari at the rate per cent, agreed on ; thus the ratio at < 3 per cent, is '03. 3t -035- 4 •04- 4t •^45- 5 -05. St '^55- 6 '06, EZAHPLB9. *"The following theorems will shew all the fiossiUe cases of limple interest, where / == principal, / s time, r s ratio, and « = amount. III. r+fzza. It. "-t-'t. a IV t^-r /r+,^^ lt6 • AtltHMBTIC EXAMPLES. ^ r. Wltattft (!te kitetast of 945L 401. lor 3 years^ at 5 per cent, per annum ? 945-5 .' 'OS 47'27S 3 141-825 20 16*500 12 , 6*000 Ans. 14 iL 16s. 6d. a. What is the interest of 796L 15s. for 5 years, at 4 per cent, per ^hnum ? Ans. 179L 5s. 4-^. 3. What is the interest of 537I. 153. from November II, 1764, to June 5, 1765, at 3-|- per cent. ? Ans. I iL --d. COMMISSION. Commission is an allowance of so much per cent, to a factor or correspondent abroad, for buying and selling goods for his employer. EXAMPLES* BROtSIUGE. 187 EXAMPLES. I. What comes the commisuon of 500L 138. 6d. toat^ 500I. 138. 6d. T . . 3t . ' ' 1502 25b 6 6 9 17-52 20 7 3 IO-47 12 4 5*67 . 4 • 2*68 Ans. 17I. I OS. 5jd. 2. My correspondent writes me word, that he h'as bought .goods on my account to the value of 754I. i68. What does his commission come to at 2|- per cent. ? Ans. 1 81. 17s'. 4-|d. , 3. What must I allow my correspondent for disbursing on my account 529L i8s. 5d. at 2^ per cent. ? Ans. nl. 1 8s. $^.^ BROKERAGE. Brokerage is an allowance of io muck per cent, to a person, culled a Broker, for assisting merchants or factors- -in- procuring or 'disposing of goods. BXAMPLfiS. l88 ' jiUTHHSTItf. SXAMPLES. J. What is the brokerage of 610L at 5$. 0|r ^ per cent*? 5s. is ^ 610L 1-52 Id 20 10*50 12 6*oo Ans. il. IDS. 6d. ■r 2. If I allow my broker 3^ per cent, what may he de- max^d, when hfr sells goods to the value of 876I. 53. lod. } Ans. 32L 17s. 2-^, 3. What is the brokerage of 879I. i8s. at |. per cent i Ans. ^L 5s. 11^ INSURANCE, fi^CRANCE is a premium of so much per cent given tp certain persons and offices for a security 6f making godX the loss of ships, houses, merchandize, &c. which may happen from storms, fire^ &c. EXAMPLBS. DISCOUNT. 1S9 EXAMPLES* I. What 18 the insurance of 874I. 138- 6d.at I3xp^ ecnt ? ' J874I. 13s. 6d. 12 ?049<5 2 874 13 6 437 6 9 Ii8'o8 2 3 ,20 1*62 12 - 7*47 4 1-88 ^' Ans. liSI. is. *j-^d. ?r 2, What is the insurance of pool, at io-|; per cent/?' t Ans. 96I. 15s. 3. What is the insurance of 1200I. at 7|- per cent. ? « Ans. 91I. 109. DISCOUNT. ji^S Discount is an allowance made for the payment: o( any $um of money before it becomes due ; and id the difier* pnce between that sum due ?ome time hence, and its pres- ent worth. The ipo AXTKVETIC. The preseni worth of any sum, or dtht, due some time hence is such a sum, as, iif put to interest, . would in that time and at the rate per cent, for which the discount is to be made, amount to the sum or debt then due. RULE.* 1. As the amount of idol, for the given rate knd time is to lool. so is the given sum or debt to the present worth. 2. Subtract the present worth from the given sum, and the remainder is the discount required. Or, As the amount of looI. for the given rate and time is to the interest of lool. for that time, so is the given sum or debt to the discount required. ^ EXAMPLES. * That an allo^snce ought to be made for paying money be- fore it becomes due> which is supposed to bear no interest till af^ ter it is due, is very reasonable ; for, if I keep the money in niy^ own hands till the debt becomes due, it is plain I may make an ' advantage of it by putting it out to interest for that time ;' but if I pay it before it is due, ii^ is giving that benefit to another ; there- ave only to enquire what discount ought to be allowed. &^ some debtors will be ready to say, that since by not ij j^iij ^ the money till it becomes due, they may employ it at in- m'esfimi^refpre by paying it before due, they shall lose that inttr- estf and fof that reason all such interest ought to be discoimt- ed ^but that is false, for * they cannot be s^d to lose that interest till the time the' debt becomes due arjives > whereas we are to consider what would properly be Idst at present, by paying die debt before it becomes due ; and this can, in point of equity or justice. EXAMPLES. I. What is the discount of 5'/3L 15s. due 3 years hence, ^ 4|- per cent. ? 4]. I OS. 3 13 10 100 ■ ■ ■ 1. $• L s. 113 10 : 13 10 : : 573 15 20" 20 20 2270 270 ^^475 270 803250 22950 .(a,o) 227,0)30^825,0(^36,4 ' 828 1472 68 4 1 105 197 12 . 227)2364(10 4 ' 376(1 Ans.^81. 4S. io|d. 149 2- yhat ]astice> be no other than such a sum^ as, being put out to interest till the, debt becomes due, would amount to the interest of the debt for the same time. — It is beside plain, that the advantage arising from discharging a debt, due some ume hence, by a pres- en^t a.. What is the present worth of 150I. payable In ^ of a year^ discount being at 5 per cent, i Ans* i4ffl[. 23. ii~d. 3. Bought a quantity of goods for 150I. ready money, and sold them again for 200I. payable at ^ of a year hence ; wh^t was the gain in ready money, supposing di^ ♦ count to be made at 5 per cent. ? Ans. 42L 15s. 5d. 4* What is the present worth of 1 20I. payable as fol- lows, viz. 50I. at 3 months, 50I; at 5 months, and the rest at 8 months, discount being at 6 per cent. ? Ans. 1 1 71. 5$. 5-^d. DISCOUNT : payrapt. ent payn^pt, according to the principles we have mentionec^ is exactly the same as employing^ the whole sum at interest till the time the debt becomes due arrives ; for if the discount allowed for present payment be put out to interest far that time, its amount will be the 'same as the interest of the v{hole debt for the same time : thus, the discount of lo^l. due one year hence, reckoning interest at 5 per cent, will be 5I. and 5I. put out to interest at 5 per cent, for one year, will amount to 5I. 5s. which is exactly equal to the interest of 105I. for one year at 5 per cent. The truth of the rule foY working is evident from the nature of simple interest : for since the debt may be considered as the amount of some principal (called here the present worth) m^a certain rate per cent, "and for the given time, that amount must be in the same proportion, either to its principal or interest, as the amount of any other sum, at the same rate, and for the same time, is to its principal or interest. DISCOUNT BT DECIMALS* ^93 DISCOUNT BY DECIMALS. RULE.* As the amount of il. for the given time is to il. so is the interest of the debt for the said time to the discount icequired. Subtract the discount from the principal, and the re- mainder will be the present worth. EXAMPLES. I. What is the discount of 573I. 15s. diie 3 years hence, at 4-j- per cent, per annum ? •045 * Let m represent any dfebt, and n the time of payment ; then will the following tables exhibit all the variety, that can happen with respect to present worth and discount. i »A Of the Present Worth of Money paid before it is due AT Simple Interest, The present worth of any sum i«. Rate per cent. Fbt n years. n months. « days. r per cent. loom «r+ioo tloom nr+ 1 200 36500m ar+ 36500 3 per cent. loom 3/1 -|- 100 400m /jt}-400 36500m 4 par cent. 25W 30OWI 9T25m «+9i25 5 per cent. 20m 240m 11+240 7300W n + 20 Of 194 ARITHMETIC. "045X3+i = i'i35= amount of il. for the gfven time. ' And S73'^75X-o45X3 = 77-45625tif interest of the debt ioT the given time. i«i35 : I : : 77*45625 : i-'35)77-45625(<58-243 6810 9356 9080 2762 2270 4925 4540 3850 3405 68'243=681. 4s. loj^d. Ans. . 445 "^2. What Of Discounts to be ali^wed for paying of Money BEFORE IT FALLS DUE AT SiMPLE INTEREST. The discount of any sum m. Rate per cent. For n years. n months. n days. r per cent. mnr nr+100 nmr mnr «r-|-i200 «r+ 36500 3 per cent. 3«+ioo flift ^mn fi+400 3«+ 36500 4 per cent. mff« mn mn n+2S «+300 fi+9125 5 per cent. tnti fHfi mn n + 20 fz+240 12+7300 EQUATION OF PAYMENTS. 195 2* What is the discount of 725). i6s. for 5 months, at 3|- per cent, per annum ? Ans. iil. los. 3~d. 3. What ready money will discharge a debt of 1377L 13s. 4d. due 2 years, 3 quarters 'and 25 day6 hence, dis- counting at 4-|- per cent, per annum ? Ans. 1226I. 8s. 84 d. EQUATION OF PAYMENTS. Equation of Payments is the finding a time to pay at once several debts, due at diiTerent tini'js, so that no loss shall be sustained by either party. RULE.* Multiply each payment by tlie time, at which it is due 5 then divide the sum of the products by the siim of the payments^ and the quotient willj)e the time required. EXAMPLES. * This rule is founded upon a supposition, that the sum of the interests of the several debts, which are payable before the equated .time, from their terms to that time, ought to be equal to the sum of the interests of the debts payable after the equated time, from that time to their terms. Among others, that^ defend this prin- ciple, Mr. Cocker endeavours to prove it td be right by this ar- gument : that what is gained by keeping some of the debts after they are due, is lost by paying others before they are due : but this cannot be the case ; for though by keeping a debt unpaid af- ter it is due there is gained the interest of it for that time,- yet by paying a debt before it is due the payer does not lose the interest for that time, but the discount only, which is less than the inter- est, and therefore the rule is not true. Although . I. 196 * ARITHMETIC- EXAMPLES. I. A owes B 190I. to be paid as follows, viz. 50I. in $ months, 60I. in 7 months, and 80I. in 10 months ; what is the equated time to pay the Vhole ? 50 X 6=300 60 X 7=420 80X10=800 50 + 60 + 80 = 190)1520(8 1520 Ansr. 8 months. .'.a; A Idiough this rule be not accurately true, yet in most questions, that occur in business, the error is so- trifling, that it will bj muc^ iised. That the rule is universally agreeable to the supposition may, be thus demonstratecl. • . . • f d z= first debt payable, and the ^stance of its term of I payment /. ^ Let \ D =1 last debt payable, and the distance of its term T. I X = distance of the equated time. (^ r = rate of interest of il. for one year. (The distance of the time t and x is = X — /. The disunce of the time T and .v is = T—x. * Now the interest of d for the time x — / is x — f X ^r | and the interest of D for the time T—x is f^xDr ; therefore x—i ^Jr=7' — ;x:.X -Or by the supposition j and frpm this equation DT+di ' X is found = JJJ^d * ^vhich is the rule. And the came migh^ be shewn of any number of payments. The true rule is given in equation of payments by decimals* EQJJATION OF PAYMENTS BY DECIMALS. * I97 2. A owes B 52I. 7s. 6d. to be paid in 4^ nionths, Sol. 10s. to be paid in 3-^ months, and 76I. 2s. 6d. to be paid ' in S months 5 what is the equated time to pay the whole ? Ans. 4 months, 8 days, 3. A owes B 240I. to be paid in 6 months, but in one month and a l^alf pays him 60]. and in 4-^ months after that 80I. more ; how much longer than 6 montlis should B in equity^ defer the rest ? Ans. 3-j?- montlm 4. A debt is to be paid as follows, viz. -^ at 2 months, ■g- at,3 months, -g- at 4 montlis, ^ at 5 months, and the rest at 7 months ; what is the equated time to pay the whole ? . ^ Ans. 4 months and 18 days. EQUATION OF PAYMENTS BY DECIMALS. ^wo delfts being due at different times^ to find the equated time to pay the nvhole^ RULE.* I. To the sum of both payments add the continual product of the first payment, the rate, pr interest of il. for bne year, and the time between the payments, and call this the first number, 2. Multiply * No rule in arithmetic has been the occasion of so many dis- putes, as that of Equation of Paymciits. Almost every water iipon this subject has endeavoured to shew the fallacy of the meth- ods made use of by other authors, and to substitute a new one in their stead. ]put the only true rule seems to be that of Mr. Malcolm, or one similar to it in its essential principles, de- rived from t!ie cousideraticn of interist and discount. The 9 *^' ^ t 19B arithmetic; 2. Multiply twice the first payment 'by the rate, and call this the second number. 3. Divide The rule, given above, is the same as Mr. Malcolm's, except ^ that it is not encumbered with the time before any payment is due> that being no necessary part of the operation. Demonstration of the Rule. Suppose a sum of money. it be due immediately, and another sum at the expiration of a ctrtain given time forward, and it is proposed to find a time ta pay the whole at once, so that neither party shall sustain loss. Now, it is plain, that the equated time must fall between those of the two payments ; and that what is got by keeping the first debt aftet it is due, should be equal to what is lost by paying the second debt before it is due. But the gain, arising from the keepbg of a sum of money after it is due, is evidently equal to the interest of the debt for that time. Ana the loss, which is -sustained by the paying of a sum of money before it is due, is evidently equal to the discoimt of the debt for that time. Therefore, it is obvious, that the debtor must retain the sum immediately due, or the first payment, till its interest shall be equal to the discount of the second sum for the time it is paid be- fore due ; because, in that case, the gain and loss will be equal, and consequently neither party can be the loser. Now, to find such a time, let a zz first payment, b = second, and / =: time between the payments ; r =: rate, or interest of il. for one year, and y = equated time after the first payment. Then arx = interest of a for x time, ^ J htr — hrx ,. /• f r 1 • and =: discount of b for the time / — x, I +/r — rx But EQJJATlbN /iF #At1I£KTS BY DECIMALS. ipj^ 3. Divide the first number by tlie second, and call the quotient the third number. 4- CaU But arx =: by the question, fi-am which cquadon * is found =t ^+^+^^^ + <L+J'+^r\\ ^P 2ar — — 2ar . I ar\ Let "* "* be put equal to n, and — = m. 2ar ar Then It is evident that «, or its equal «*|* is greater than «* — »ii*, and therefore x will have two affirmative values, the ^antities «+»* — ^^^wi|* and n — 11 * — ifi| being both positive. But only one of those values will answer the conditions of the question ; and, in all cases of this problem, x will be sr n — • . X For suppose the contrary,, and \tx.xzzn +«* — «!*. Then / — *?=/ — B— «* — m]*=/ — r^\ — Ji* — ml*=- T X X X Now, since a+b+atr X — =/»> and & X — = «f we sfiall 2^ir have from the first of these equations ,/* — 2/«= — ^/— c/ X — and consequently /— x= n* — bt-^at X — I — «* — ^X- — ^ j ar\ Bat aoo . ARlTHMiTlC. . . : - 4* Call the $quare of the'thitd number the fourth num-> bcr. g. Divide the product of the second payment, and time between the payments, by the product of the first pay- ment and the rate, and call the quotient the fifth number. '6^ Txoni the foiurth numbev take the fifth, and call the square root of the difference the sixth number. 7. Then the difference of the third and sixth numbers is the equated timey after the first payment is due, EXAMPLES. I. There is ioot. payable one year herfce, aftd 105I. payable 3 years hence 5 what is the equated time, allow-* ing simple interest at 5 per cent, per annum ? TOO Is But n * — 5i X " is evidently greater than 1\^ TF and therefore «* — Lt >( — — ^/i* — k — ai^ — f , or itsp equai f — X, must be a negat Ire quantity ; and consequently x will btf greater tlian, /, tliat is, the equated time will fall beyond the sec- ond payment, which is absurd. The value of *r, therefore, cau- not be = -3—1- — + — i—X 2ar zar ar X , but must in all cases "" 2ar lar \ ar which h tlie sam« as the rule. From this it Ippears, that the double sigh made use of by Mr. Malcolm, and every author since, who has given his method, cannot obtain, and that there is 'no ambiguity in the problem. In like manner it might be shewn, that the directions, usually given for finding tlie equated time when there arc more than two paymentf; EQUATION OF^tATMSNTi BY DECIMALS. 201' lOO 100 . •05 "1 S'oo 200 2 'OS lO'OO lo'oQsad numi 100 • 105 ' ^ [0)215=: ist number. 2 1 -5 = 3d number. 21-5 1075 \ 215 430 ^ iljen 462*25=: 4th number. 105 2 , 2st payment X rate :r'*5)2io ^2= 5 til number. ♦ 462*25 42 21-5 ( 20* 5 =2 6th number. 42o-25(2o-5 4 I =r equated time from the first payment, and .'. 2 years K whole, equated time. 405)2025 2025 » 2. Suppose payments, will not agree with the hypothesis, but this may be easily seen by worldng ai) example at large, and examining the truth of - the conclasion. The - •* ■ *^ 20Z AMTHMETIC. .;-, i 2. Suppose 4oaL ipe to be paid at the end of 2 years, and 2100L at tbc'cod of 8 years s'what is the equated time for one payment^ reckoning 5 per cen^ simple inter- est ? ^ Ans. 7 years. 3. Suppose 3boL are to be paid at one year's end, and 300I. flUW;»r the end of " i-j- year ; it is required to find the tiiiie 1U) pay it at one payment, 5 per cent, simple intetest being allowed. ' Ans. 1*248637 year. COMPOUND INTEREST. Compound Interest is that, which arises from the prfticipal and interest taken together, as it becomes diie, at the end of each stated time of payment. I. Find the amount of the given principal, for the time of the 'first payment by simple interest. ^ z. Consider The equated time for any number of payments may be readily found when the question is proposed in numbers, but it would not be easy to give algebraic theorems for those cases, ^ on account of the variation of the debts and times, and the difficulty of finding between which of the payments the equated time would happenl Supposing r to be the amount of il. for one year, and the oth- cr letters as before, then / — s will be a general theorem log, r . for the equated time of any two payments, reckoning compound interest, and is found in the same manner as the former. * The reason of this rule is evident from the definition, and the principles of simple interest. . c eOMPOUND INTEHI^T. „^ 203 2. Conner this amount as the pritibipal for the second payment, whose amount calculate as bSCorei and so on through all the payments to the last, still jw:countii)g the- last amount as the principal for the next payment. • ^. ' EXAMPLES. I. What 19 the amount of 320I. los. for 4 years> jt 5 To)32ol. 16 I OS. 6d. kAAL/%^ »*«««• ««*«iV>« W>U»i ■ I St year's principal. I St year's interest ad year's principaL 2d year's interest. 3d year's principaL 3d year's interest. 4th year's principal. 4th year's interest. whole amount, or the ^ T?)836 16 10 16 6 6i A)3S3 ^7 7 13 4 18 II 389 quired. . II 4t answer re- 2. .What is the compound interest of 760I. los. forborn 4 years at 4 per cent, i Ans. 129I. 3s. (S~d. 3. What is the compound interest of 410I. forborn for 2-j- years, at 4- per cent, per annum 5 the interest payable half-yearly ? Ans. 48I. 4s. .ii-|d. 4. Find the several amounts of 50!. payable yearly, lialf-yearly and quarterly, being forborn 5 years, at 5 per cent, per annum, compound interest. Ans. 63I. 163. 3-^d. 64I. and'641. IS. 9^d. ' COMPOUND COMPOUN^ IJ5TEREST BY DECIMALS. .' V^** RULE.* I. Fifld^l^^mount of il. for one year at the given rate percent.:.:?;' 2. Involve J^.a|||^monstration. Let r = amount of il. for one year, ^d.M=: principal or given sum ; then since r is the amount of il.^ror one year, r* will be its amount for two years, r' for 3 years, and so on ; for, when the rate and time are the same, all principal sums are necessarily as their amounts ; and consequently as r is the principal for the second year, it will be as i : r : : r : r*z: amount for the second year, or principal for the t)bird ; and again, as i : r :z r* ; r'= amount for the third year, or prin- cipal for the fourth, and so on to any number of years. And if the number of years be denoted by /, the amount of il. for < years will be A Hence it will appear, that the amount of any other principal sum f for / years is pr^ ; for as'; ; r* : ; / : p/t the same as in the rule* If the rate of interest be determined to any otlier time than a year, as i, j^f &c. the rule is the same, and then / will represent that stated time. 'fr:=z amount of il. for one year, at the given rat^ per cent. f zz principal, or sum put out to interest. / z= interest. t = time. ^ m = amount for the time /. Then the following theorems will exhibit the solutions of aU the cases in compound interest. I. fr^zzm. IL prf—pz=:u Let IILi^=/. IV. Jli =r. The /COMFOUND I-NTEKEST B? DECI TRIALS. ^ 2^5 2^ Involve the ana bint thus found to such a power, as ii denoted by the number of yeurs. 3. Multiply this power by the prindpal, or given sum, and the product will be the amount requircdi *. y|» Subtract : ■ - - - ■ ■ ■ - ■ ■ • ■» r - J The most c©nvenient way of giving the theorem for theVm^, as well as for all the other cases, will be by logarithms^ as fallows : I. / X %. r-^log. p-=.log. m, II. iog, m — t X log, rzra^f^ HI. '^-fi:%^L=/. Jog, r t If the compound interest, or amount of any sum, be required for the parts of a year, it may be determined as follows : I. Whsn the time is any aliquot part of a year. RULE. 1. Find the amount of il. for one year, as before, and that root of it, which is denoted by the aliquot part, will be the amount sought. * ^ 2. Multiply the amount thus found by the principal, and it will J^ the amount of the given sum required. II. fVhen the time is not an aliquot part of a year. RULE. I. Reduce the time into days, and the 365th root of the jimount of il. for one year is the amount for one day. % 2. Raise this a:r.ount to that power, whose index is equal to ' the number of days, and it will be the amount of il. for the giv- en time. 3. Multiply this amount by the principal, and \\ \viB be the . amount of the given sum required. To avoid extracting very high roots, the sam.e may be done. by logarithms thus : divide the logarithm of the rate, or amount of il. for one year, by the denominator of the given aliquot part, * aud the quotient will ba the logarithm of the root so-.ight. 206 ARITHMETIC 4. Subtract the principal from the amount, and the re^ mainder will be the interest. (EXAMPLES. I. What is the compound interest of 500L for 4 yearsj at 5 per cent, per annum i 1*05 = amount of iL for One year at 5 1*05 per cent. . ..- " 525 ^ 1050' I'I025 II02J 55125 22050 1 10250 1 1025 i'2 1 550625 zz4th power of 1*05. 500Z1: principal. 607*753 12500:= amount. 500 io7'753i25 zi 107I. 15s. o|<l. rr interest required* 2. What is the amount of 760I. los. for 4 years> at 4 per cent. ? Ans. 889I. 13s. 6^d. 3. What is the amount of 72 il. for 21 years, at 4 per cent, per annum ? Ans. 1642I. 19s. iod« 4. What is the amount of 217L forborn 2~ years, at 5 per cent, per annum, supposing the interest payable quar- terly ? Ans. 242I. 13s. 4-|<l. ANNUITIES. ANNUITIES. 207 ANNUITIES. An Annuity, is a sum of money payable every year, for a certain number of years, or for ever. When the debtor keeps the annuity in his own hands^ beyond the time of payment, it is said to be in arrears. The sum of all the annuities for ttie time they have been forborn, togethfo with the" interest due upon each, is called the amount. If an annuity be to be bought off, or paid all at once^ at the beginning of the first year, the ^rice, which ougnt to be given for iti is called the present ivorth. To find the Amount of an Annuity at Simple Interest. RULE.* 1. Find the sum of the natural series of numbers i, 2, 3, &c. to the number of years less one* 2. Multiply — ..I — I — 4 ■■ . . - - * Demonstration. Whatever the time is, there is due up- on the first year's annuity, as many years' interest as the whole number of years less one ; and gradually one less upon every suc- ceeding year to the last but one ; upon which there is due only one year's interest^ and none upon the last ; therefore in the whole there is due as many years' interest of the annuity, as the sum of the series i> 2> 3> 4> &c. to the number of years less one. Consequently one year's interest, multiplied by this sum, must be the whole interest due ; to which if all the annuities be added, the sum is plainly the amount. Q^ £.. D. • Let r be the ratio, n the annuity, t the time, and a the amount. Then will the following theorems give the solutions of all the different cases. T t^m — im ,. rr ^^ — 2/11 I. ■ ■ ■ +tnzsa. II. — T —r. 2 t^n-^tn u 208 ARlTHMfiTfCi 2. Multiply this suiji by one year/s interest of the an- nuity, and tins product v.-ill be tkc-whoWnnterest due upon the annuity. 3. To this product add tile product of tte annuity and time, and the sum will be th^ amount sdilghl:. Note. Whca t^c annuity is to be paid half-yearly or quarterly ; then take, in the former case, 4- the ratio, half the annuity^ and twice the number of years j and, in the letter ca^e, ^ the ratio, ~ the annuity, and 4 times the number of years, and proceed as before. ^ . EXAMPLES. I. What is the amount of an annuity of 50}. for 7' years, allowing simple interest at 5 per cent. ? 1+2+3+4+5+6=21=3X7 2I. 10s. zz I year's interest of 5^!. 3 7 10 7 52 10 • 350 o = 50I. X 7 402I. I OS. = amount required. 2, If -X . ■ =;/. IV. 1 — I Ez^ /V — /r+2/ '^41 ^ In the iast theorem J= — ZI—, and in theorem fir^, if a ««• m cannot be found equal to the amount, the problem is impossible in whole years. ' Note. Some writers look upon this method of finding the amount of an annuity as a species of compound interest ; the annui- ty itself, they say, being properly the pimple interest, and the cap- ital, whence it arises, the principal. ANNUITIES. 209 li- on a. If a pension of 600I. per annum be forborn 5 years, •what will it amount to, allowing 4 per cent, simple in- terest ? Ans. 32iijol. J. What will an annuity of 250I. amount to in .7 years, to be paid by half-yearly payments, at 6 pef cent, per an- ^^1 Hum, simple* interest i . > Ans. 209 il. 55^ nd tile tie ^ To'Jind the present Worth of an Annuity of Simple Interest* RUL£.* Find the present worth of each year by itself, discount- ^^ !fig from the time it becomes due, and the sum of all these will be the present worth required* EXAMPLES. . * The rcasoii of this rule is manifest from the nature of discount^ for all the annuities may be considered separately, as so many sin»- l^d and indepebdetit debts, due after i, 2, 3, &c. years ; so that the present worth of each being found, their sum must be the present worth of the whole. Hie estimation j however, of annuities at simple interest is high- ly unreasonable and absurd. One instance only will be sufficient - to shew the truth of this assertion. The price of an annuity of 50I. to continue 40 years, discounting at 5 per cent, "^ill, by ei- ther of the rvtfes, amount to a sum, of which one* year's interest only exceeds the annuity* Would it not ^therefore be Highly ri*- diculous to give, for an annuity to continue only 40 years, a snixiy which would yield a greater yearly interest for ever ? It is most equitable to allow compound interest. Let p =: present worth, and the other letters as before. Then ^ fzX— i" + — — +— i— .Scc-tO' * iJ^r l-}-2r i+3r l-f-/r i * t ,1 o "^^ I p-^. 1- — y {-■■■. , Scclo .H - ■'■» =«' C c The 210 . . ARITHMETIC. ElfAMPLES. . i 1. What is the present worth of an annuity of lool. tor continue 5 years, at 6 per cent, pcramxum, simple interest ? 106 : 100 :: xoo : 94*3396 = prcfcnt worth for i yean 112 : 100 :: 100 : 89*2857 =i 2d year. 118 : 100 : : 100 : 84*7457 =« 3^ jcar.^ 124 : 100 :: *ioo : 8o'645i = 4th year* I JO : 100 :: 100 : 76*9230= 5th year. 425-9591 = 425L i8s- 9^d. = presenlt ^j worth of the annuity required. 2. What is the present worth of an annuity or pension of 500L to continue 4 ywi'rs, at 5 per cent, per annum^ . simple interest i Ans« 1782L 5s^ ^ii To find the Amount of an Annuity at Compound Interest* RULE.* I. Make i the first term of a geometrical ptogressioi^ and the- amount of iL for one year, at the given rate per cent, the ratio. ^ Carrf The other two theorems for the time and rate cannot be give* in general terms. * Demonstratiok. It is pWn, that upon the first year's an- nuity, there will be due as many years* compound interest, as the ^▼cn number of years less one, and gradually one year's interest less ANNUITIES. 211 a. Carry the scries to as many terms as the number of *|j? years, and find its sum. 2r Multiply tKc'sum thus found bj the given ainnUity, and the product ViH be the aniount sought. EXAMPLES. less upon every succeeding year to that preceding the last, which has but one year's interest, and the last bears no interest. Letr,therefore, = rate,or amount of il.for i year; then the series , i of amounts of |1. annuity, for several years, from the first to the last, is I, r, r*, rS &c. to r'"'-. And the sum of this, accotding to the rule lu geometrical progression, will be — — , = amount of fL anni;tity (ot t years* And aU annuities are proportional to , . t ' r ' ' ^t — ^ ^ — I their amounts, therefore i : ■■ : ; « : X « = r — I r — I- amount of any given annuity n, Qi,E. D. . Let r IT rate, or amount of il. for one year, and the other letters as before, then X«^^> and =«. r— I r' — I % And from these equadons all the cases relating to annuities, or* pensions in arrears, may be conveniently exhibited in logarithmic t|rms, thus s I. Log.U'i^Loi. r* — ^^i 1 — Li^. r — i z^Log. a, II. Log* a-^Log* if — ^i *^Xj)g. r'^i:=zLog, «. Log. r. n n 212 ARITHMETIC. e3:ampl£s. I. What is the amount of an anhuity; of 40!. tp con«# tinue 5 years, "aHowing 5 per cent^, cojnpibi^id iftteres£ ? ' ^ + i*95+i'o5| +i-os|+ro5|= 5-52563125 5-52563125 40 . 2^ro2525 ■' '• ' '26 ■'' ■ ■ ; • ■ > -.Si iz ; ; 0-505 ■•"■ 12- - . . # ^•06 ' Ahs^ 22rii-64; i ' 2. If 5 ol. yearly reflt, •oi' annuity, be; fdrborn 7 ycar% what will it amount to, at 4 per cent, per annum, com- pound intcr<?st ? '^ • '■ • ' —'' Ans. 395!% To find the present Value of Annuities at Cotopotmd Interest. * RULE.*. I. Divide the annuity by the ratio, or the amount of jl. for Q{ie year, and the quotient will be the jwresent M'orth of I year's annuity. '•' ■• 2. DiviSe * The reasQii of this rule "is evident firom; the nature of the question, and what was said upon tlie same subject in the purchaS'* ing of annuities at simple interest. I^et p = present worth of the annuhy, and the other letters as bcTore, then as the amount =r -X'^j and as the present wortk i 'ANNUITIES. llqr 2. Divide the annuity by the square of the ratio, and tbc^tient will be th6 pre&eut wortb of. the ^jonuity for Itwo.years. .1 =■ » '. ■; ;■ •-■: ^:::. . -I :..:.:'.•> 3. Find, in like manner, the present worth of •"•each, yeat hy itself,:, and the sum of .,»U. these will be the value, of the anniiity sought. - .1 '• \;.'. fA ■■•". .-/■' ■<_■-■■ ■ ^ :EXA-MPLEft ... y - m .^ m u ■ , « , hm , "^ or principal of this, according to the principles of compound in- tcrcstj is the amount divided -by ,rf^ therefore - «And from these theorem? all. the cases, where the purchase of * annuities is concerned, may be eXhi})ited in logarithmic terms, as follows : I. Log.n^Log,! jLog.r — iziLog.p, 11. Log^p-\'Log. r — I — Log, I — — -zzLog. n. HI. i^ir:£si!!+^=^=/. iv, r^-'-^+ixr^+^'^o. Log.r P p Let / express the number of half years or quarters, n the half year's or quarter's payment, and r the sum of one pound and \ or J year's interest, then all the preceding rules are applicable to half-yearly and quarterly payments, the same as to whole years. ySff amount of an annuity may also he found for years And parts of • ' a year^ thus : 1. Find the amount for the whole years as before. 2. Find the interest of that amount for the given parts of a year. 3. Add this intetot to the former account, and it will give thfe whole amount required. - rh(k 214 ARITHMETIC* EXAMPLES. i^ What is the present worth, of an annuity of 40!. t» continue 5 years, discounting at 5 per cent, per ann^n^ compound interest ? . ratio = i*05)4O'0oooo(38-o95=presenttir0rthftJrxyeaf, #atioj=- i'io25)4o'ooooo(36'28i= do. for 2 years, ratiofir ri57525)4o-o6ooo(34-556== do. for 3 years, ratio j = r2i55o6)40'ooooQ(3a(f899=r don for 4ycarg.f ratio] = i'276ai8)40*ocooo(3i'342=: . do.' for 5 years, whole present worth of the annuity required. 2. What is the present worth of an annuity of axL I OS. 9-^d. to continue 7 years, at 6 per cent, per annum^ (Compound interest ? Ans. 120L 5s. 3. What is 70I. per annum, to continue 59 years, worth in present money, at the rate of 5 per cent, per' annum ? * Ans. X32i*302iL The present worth of an annuity for years and parts of a year matf he found, thus : 1. Find the present worth for the whole years as before. 2. Find the present worth of this present worth, discountiiig for the given parts of a year, and it will be the whole present Worth required. to Jini the prtseni Worth of a Freehold Estate^ or an ,Anma^ i^jQ continue for ever, at -Compound Interest. HULE.* As the rate per c^t is to lool. so U the yearljr rent tcf the value required. * The reason of this rule is obvious : for since a year's interest j|., of the price, which is given for it, is the annuity, there can neither more nor less be made of that price than of the annuity, whether it be employed at simple or compound interest. The same thing may he shewn thus : the present worth of an tunuity to continue forever is JI4. ^4. Jl+i, &q. 4d ufrntum, r r* r^ r^ iB has been shewn before } but the sum of this series, by the rules of geometrical progression, is — 2— $ tberefbre r-^-x -t 1 « « « i r— I «p^ 9 which is the rule* #— I The following theorems shew all the varieties of this rule. 1. -JL=/. IL i^ X/=«. HI. ^ + i=r, or-=r— .1. r— I p . t The price of a freehold estate, or annuity fo continue fcr €ver, at simple interest, would be expressed by -JL, + — '^ . i+r ^ i + 2r +- , Stc. ad hfotiom ; but the suili ortfaas se- ries is infinite, or greater than any assignable number, which sufficiently she^ws the absurdity of using simple ioterestiki these cases. \ XlC' AElTHMETXiC. EXAMPLES. L An.estatc brings in yearly 79I. 4s.'*wfeaf wdtlU it «cfl for, allowing the purchaser 4^ per cent, compound interest for his money ? . . 4*5 : 100 : : 79*2 i '• • * •'- •• ■• ' t6o' • , •' ■• 4'5)7920'o(i76oL the answer^ 45 ' • 270 270 ■ 4. "What is the price of ^ perpetual annuity of 40I.. dis- counting at 5 per cent, compound interest ? Ans. 800L 3. What 18 a freehold estate of 75I. a year worth, al- lowing the buyer 6 per cent, compound interest for his money ? Ans. 1250L To find the present Worth of an Annmijj or Preehold Estate^ in Reversion^ at Compound Interest. RULE.* I. Find the present worth of the annuity, as if it*were to be. entered on immediatcly- 2. Find * This rule is sufficiently evident without a demonstration. ThosCy who wish to be acquainted with the manner of compil- ing the values of annuities upon liyes, may consult the writings of Mr. Demoivre, Mr. Simpson, and Dr. Price, all of whom have handled this subject in a very skilful and mast;erly manner. Dr. ANNTTITIES* 217 2, Find the present worth of the "last present worth, ^ discounting for the time between the purchase and com- mencement of the annuity, and it will be the answer t€* quired. EXAMPLE^i X. The reversion of a freehold estate of 79L 4s. per aii- num, to commehc6 7 years hence, is to be sold j what is It worth in ready money, allowing the purchaser* 4— pet fccnt. for his money ? 4'5 : 100 : : 79-2 ibo 4'5)792o'#(i^6oi= pfescnt worthy 45 if. entered on im^ — i— * mediately* 31S 276* and ro45J = i*36o862)i76o'ooo(i293'a97 = 1293I. 5$J ii^d. = present worth of 1760L for 7 years, or the whofe present worth required; ^ • 2. Which is most advantageous, a term of 15 years fa an estate of lool. per annum, or the reversion of such an estate forever, after the expiration of the said 15 years, computing Dr. Price's Tre^use upofi Annuities and Reversionary Pay- ments is an excellent performance, and will be found a very val« ] nable acquisition to those, whose incUoAtions lead them to sudi<» •f this nature. D B il8 ARITHMETIC. '. computing at the rate*of 5 per cent, per annum, compound intctet ? ^ Ans. The first term of 15 years js better than the re- version for ever afterward, by 75I. 18s. 7-jJ. 3. Suppose I would add 5 years ta a running lease of .15 years to come, the improved rent being 186I. 7s. 6d. per annum \ what ought I to pay down for this favour, ^discounting at 4 per cent, per am^um, compound interest I Ans* 460L 14s. i^» -7^ P0SITION. Position is a method of performing such questions, as cannot be resolved by the common direct rules, and is of two kuids, called single and double. Zingle Position teaches to resolve those questions, whose results are proportional to their suppositions. RULE.* 1. Take any number and perform the same operations^ ^with it, as arc described to be performed in the question. 2. Then say, as the result of the operation is to the po- sition, so is die result in the question to the number re- quired. EXAMPLES^ * Such questions properly belong to this rule, as require the multiplication or division of the number sought by any proposed number ; or when it is to be increased or diminished by itself, or any part»^ of itself, a certain proposed number of times. For m this rosiTioN. 219 EXAMPLES. X- A's age is douMe that of B, apd B's is triple that of ^ Cy and the sum of all their ages is 140 : what is each person's age ? , ^ Suppose A*s age to be 60, Then will B's = V =30* AndCs = V = lo- 100 s,um. As 100 : 60 : : 140 : IJij5ii|-:^ =84= A's age. Consequently .V s=42 = B's. And V =i4=Cs. 140 Proof. 2. A certain sum of mpney is to be divided between 4 persons^ in such a manner, that the first shall have y of it ; ' the second ~ j the third :y 1 said th^foi^ the remainder, which is 28L -: wfcat k the sum i ^ Ans. 11 21^, 3. A person, after spending -5- and ^ of his money, had 60I. left : what had he at first ? Ans. lA^l. 4. What number is that, which being increased by -1^ y and ^ of itself, the sum shall be 125 ? A^. 60. this case the reason of the rule is ol^ious ; it ^.i^ tlien evident, that the results are proportional t<f the suppositions. fix I X : : na : a f : « I : t . a ■ Thus <{ » n • f +f-,&c. n — w : re ^« ±+±,6cc. n-^ tn : <7,6csood . Note, i may be made a constant supposition in all questions ; and in most cases it is better than any other number. '• 220 ARITHMETIC. 4 / . 5.«A. person bought a chaise, horse and harness, for (Sol. ; the horse came to twice the price of the harness, and the chaise to twice thq price of the horse and harness : what did he give fot each ? Ans. 13L 6s. 8d.-for the horse, 61. 13s. 4d. for the har- ness, and 40I. for the chaise. 6. A vessel has three cocks. A, B and C 5 A can fill it in I houry B in 2, and C in 3 : in what time will they aU fiU it together ? Ans. -—- hour. DOUBLE POSITION- Double Posiiion teaches to resolve questions by making (wo suppositions of false numbers* RULE.* I. Take any two* convenient n^imbcrs, and proceed with' each according to the conditions of the question. 2. Find * The rule is founded qn thi? si^ppo^tiop, that the first errq^ if to th^ second, as the difference between the true and first sup. posed number is to the difference between the true and second supposed number : when that is not the case, the exact answer ta the question cannot be' found by this ride. That the rule is true, according to the supposition, may be thus demonstrated. Let A and B be any two numbers, produc^ed from a and b by similar operations ; it is required to find the number, from wbi^h N is produced by a like operation. Put SQ =: number required, and let N — ^=r, and N — -5=x. . .Then, according to the supposition, on which the rule is foupd- ed, r : / : : x— *a : x~^, whence, by multiplying means and extremes. POSITION. .221 2. Find how much the results are different from the re- sult in the question. 3. Multiply each of the errors by^ the contrary supposi- jtion, and find the sum and difference of the products. 4. If the errors be alike, divide the difference of the products by the difference of tfec errors, and the quotient will be the answer. 5. If the errors be unlike, divide the sum of the prod* ucts by the sum of the errors, and the quotient will be the answer. Note. • The errors are said to be alihy when th^y are both too great or both too little ; arid tmlihy when one is . too great and the other too littje. EXAMPJLES. T' ■. . , - I. A lady bought tabby at 48. a yard, and Persian at '28. a yard 5 the whole nupiber. of yards she bought was 8, and the whole price 20s. : how many yards had she of each sort ? Suppose extremes, rx — rhzzsx — sa ; and, by transposition, rx — sx^ziri'^sa ; and, by division, xz=. • == number sought. Again, if r anA J be bodi negative, we ftadl have — r : .—1 : : X — a : x — h, and therefore — rx^^-rbzzz'^sx'^sa ; and rjc^— 4x=:rb — sa ; vbepcc a^ =: ^ as before. In like manner, if r ^r j be negative, we shall have x=s — -—-^ by working as before, which is tlie ipk. Note. It will be often advantageous to make i and o the ^uppositiojis. -:2 ARITHMETIC. Suppose 4 yards of tabby, value i6s. Then she must liave 4 yards of Persian^ value 8 ^um.of their values 24 So that Ae first error is -f" 4 Again, suppose she had 3 yards of tabby at i as. Then she mast have 5 yards of Persian at 10 Sum of their values 22 So that the second error is -J" ^ Then 4— 2n23i difFerence of the errors. Also 4X2Z1811: product of the first supposition and second error. And 3X4=112=: product of the second supposition by the first error. And 12— 8zr4zr their difference. Whence 4 — 2=12=: yards of tabby, 7 < And 8-^2=6= yards of Persian, $ '^''^^'' 2. Two persons, A and B, have both the same income ^ A saves — of his yearly 5 but B, by spending 50I. per ^. num more than A, at the end of 4 years finds himself xool. in debt : what is their income, and what do they spend per annum ? Ans. Their income is 125I. per annum; A spends lool. and B 150I. per annum. • 3. Twp persons, A and B, lay out equal sums^of money in trade ; A gains *261. and B loses 87I. and A's money is now double that of B : what did each lay out ? Ans. 300L ^. A labourer was hired fbr 40 days, upon this condi- tion*, that he should ree^ilgj^ 2od. for every day he wrought, and forfeit lod. for every day he was idle ; now he receiv- ed PERMUTATION AND COMBINATION. 223 cd at last 2I. IS. 8d. : how> many days did he work^ and how many was he idle ? ' ' Ans. He wrought 30 days^ aiid was idle lo. 5. A gcAleman has two horses of considerable value, and a saddle worth 50L j dbw, if the saddle be put on the back of the first horse, it wil) make his value double that of the second ; but if it be put on the back of the sec- ond, it will make his value triple that of the first : what is the value of each horse ? Ans. One 30I. and the other 40I. 6. There is a fish, whose head is 9 inches long, and hii tail is as long as his head and half as long a^ his body, and his body is as long as his tail and his head ; what is the whole length of the fish ? Ans. 3 feet. PERMUTATION and CpMBINATION. The Permutation of Stnantities is the shewing how many different ways the order or position of any given number of things may be changed. This is also called Variation, Ahernation^ or Changes s and the only thing to be regarded here is the order they stand in ; for no two parcels are to hare all their quanti^ ties placed in the same situation. ' . The Combination of Quantities is the shewing how often a less number of things can be taken out of a greater, and combined together, without considering their places, or the order they stand in. This is sometimes called Election, or ChoUe : and here every parcel must be different from all the rest, and no two jm are to have precisely the same quantities, or things. ^^ The CompojitioH of ^antities l|;.the taking a given num- ber of quantities out of as manf equal rows of different quantities, 224 ARITHMETIC. Mt q^uantitiesj one out of each row^ and combining them to^ gethcr. Here no regard is had to their places ; and it differs from combination only^ as that admits of but one row, or set, of things. . ^ Combinations of the snmi j^fm are those, in which there is the same number of quantities, and the same repeti- tions : thuSy ahcc^ bbadj deefy &c. are of the same form \ but abhc^ abbb^ aaccy &c. are of different forms. ^0 find the murnher of permutations ^ or changes ^ that can be made of any given number of things^ all different from 9ach other* kULEi* Multiply all the teryms of the natural series of numbers^ from I up to the given number, continually together, and the last product will be the answer required. BXAMPLEf. * The reason of the role may be shewn thus : any one thin^ a is capable on]y of one position, as a* Any two thingsj a and ^, are only capable of two tafiations \ ^ abf ba ; whose number is expressed by i X 3* If there be 3 things, «, b arid c, then any two of thenl, leaving out the tliird, will have 1X2 variations ; sfnd consequently, whdn the third is taken in, there will be 1X2X3 variations. In the same manner, when there are 4 things, every 3, leaving cut the fourth, will have 1X2X3 variations. Then, the fourth being taken in, there will be* 1X2x3X4 variations* And so on, as far a$ you please* nRMUTlTIOM AMD COMBIMATlOll. 225 exaufle;s. I. £[ow many changes may be made with the^ three lkttcr$, ahci 1 2 \ 6 the answer. « .. 1 • , 3 6 The changes. ahc - ach lac hca cab cba . 2. How n^any changes may be rung on 6 bells j^ Ans. 7io.' 3. l^or how many ^ayi can 7 persons t>e placed in a dif<* ferent position ^t dinner i Ans. §640 dayd* 4* How rtl^ttiy changed mdy be futlg oti 12 bell^i and how long Would they be in ringing, supposing 10 changed to be rung in i minute^ and the year to Consist of ^6$ days^ 5 houirs and 4^ nUnutes ? Ans. 479001600 changes^ slnd ply. a6d. litah. 4inL 5* How many changes may be made of the words in the following versie ? Tot tihl sunt dutis^ virgo, ^uat sydera Alls. 40320* vyV YEOBLKM 1 1 226 ' llttrrHllETIC. ^ PitCBliicM ll. > J^ny numhr ef Afferent thingr being gtveH, to Jlfid honu tnam changes can be made out of tiem, by toting any ^veHnum* her at a time. ^ • ... Take a series of numtjers, beginning at the number of things given, and decreasing by i till the number of terms be equal to the number of things to be taken at a time, and the product of all the terms will be the answer re- quiired. ' EXAMPLES. * This rule, expressed in tennsy is as follows : rnXm — i X m — % X m, — 3, &c. to n terms ; wkere m =: number of things i^Ten, and n = quantities to be taken at a time. In order to demonstrate the rule, it will be iici;essary^ to pre* mise the following LEMMA. " The number of changes of m things, ^^ 'I ^^ ^ tis^ 19 ^.W) to m changes of m — i things, taken »— i at a time. Demonstration. Let^any 5 qoantides, o^dlfy be given. First, leavp out die a^ a;id ^t v ir 'QUjqb(9i..Q£^the,nfia^on9 of every two, &, hd^ &c. that can be tj^j^.qfit 9C the 4, ^^ii|j4^ iog quantities, i^^d^. . . «., v Now let a be put in the first place, of ^^\^v!^^^i;^y/^ txj^ and the number of changes will stillremain.the..saQ)e i that' is, V =z number of variations of every 3 out of the 5, ahcde^ when a is first. , • . , In like manner, if ^, c^ dy e, be Sfcqevitely Icift out^ ^<iPvnr bcr.of variations of all the twos will also = v ^ and K c^ ^^ being respectively put in the first place, to make 3 quantities out of 5, there will still be v variations as before. But these are all the variations, that can happen of 3 things out of 5, when a, bj c, df e, are successively put first ^ and therefore the PBRMUTATION AUD COMBINATION. aaj EXAMPLES. . I. liow many changes may be made out of the 3 letters ^c^ &y taking 2 at a time* ' ' ' ' Or 5)^2=6' ffeaiisWei 2 Or^3Xi= 6 ' . .' . i. The'cKanges. .^ , ttC ■ - - .. « ca ■■• ■■■■=■ tb ie; H«Fi^'in^ ircards can be ma^e wfth $ letters^ of the alphabet, it bein^ admitted t^at a huixiber of consonants msf m&ea'Vorl ?■' '*' '•"•■' '' ■ Aiss:' yfsaHga • '• •• ■'.■■•■■ -'^i "-.^ '':-\ua iu s^'a, •^ PROBLEM Ac ialiiorall'tfiese is tlie 'satti of all die changes of ^ ttuAJj^'out But the iBthh'of these b. so many times v, as is the oumber of * * ■• ^ . ■ • • V • . • ." '■•■-' . J. thbgs $ thit is^ 5v» or mv» = all the changes of 3 things out of £., And the same way o^^rfaso^fg ipay be-applied t^ l^ly nam- J^,vh$teYcr. , .-i;]!; .;,,?. . • .c^;. .-: DBMottsirttAtfON OF TBfi Itu|.B. . Let any 7^ thiqg;s, ^^4ffit be g^en, and let 3 be the nomber of quantities to be taken. -'%fcai^7;Mii=^: "' ■ ' ' '^ .;;. : • -'■ , jNoV'itis evident, that the number of changes, that c^r be ipade l^y ta^ng .1 by i out of 5 things, will be 5, which let =:v« Then, by the lemma, when m=6 and n=i, the^ number of changes will :rzmvz=:6 X 5 ; whiclT let =:v a seco^id dme. . A^ain by lemma, when mzz'j and 71=3, the number of changes s±i»i?i±7X^X5-; that i^ nnic=«x«t<^i>i>n«--^9 coxitinaedto 3i or n- termsi> Atfd the sacme niay he.dt^wn for any o^ SMOBibers* r- 42* ' " AtiTHmric. PROBL£M III. jttty mtmtir rf things being givm, mAeriof there ate several given things of one sort, several of another, \^c. to find how mafPf changes can be made. out of them alL I. Take tlie series i, 4, 3, 4, &c. up to tl^e number of things givenj and find the^roduct of. all the terms. 2. Take ,; „ 1 * This rale is expressed 10 ^erms thus ; I X a X 3»&c-to/X ' X 2 X 3>&C. to y,&c. of things giien^./ = number of things of the first sort, ^?: quiy- ber of thingB of the second sort, &c» The DEMOKSTRATiOH may be shewn as follows : Apy two c^o^ntities, at h both differenty fidmh of a- chtmges ; but if the quantities be the^ same, 01 ah become aa, there will be but one alternation ; which may be expressed by ^* a=i. Any threfe quantities, abcy all different fiom each other, afibrd 6 variations ; but if the quantities be all alike, or abc become aeut^ then the 6 vaiiations will be reduced tb i •; which fliay be ex- pressed by JL2ii2ii=:i. Again, if twp of the, qofmtities only ^ 1x2x3 . ^ / be alike, or abc become aUcy then the six variations will *be re- duced to these 3, aac^ caa and oca \ which may be expressed byii<i2<3=5, ' - 1X2 Any four quantities, ahcd^ all diScrent from each Other, will admit of 24 variauons ; but if the quantities be the sapie, Q^ abed become aaaa% the number of variations will be reduced to one I pE&MtrrATi6K AND xxjUbinition. itf i. Take the series i, 2, 3, 4, A:c. up to tKc number of given tilings of the first sort, and the' series I9 0, 3^ 4*'^ &C. up ..to the; nymber of given things of the second sort, &c. ' . . ; ..; ^:, v= '':]-' '- I . 3:^;I)ivide the prodiftt of ail the terms of the first to^* ries by the joint prdduct of all the terms of the remaining ones, and the quotieot will be the answer required. * \' -. ■.'■.■.- BX-AMPLES. ' ■ ■■.'.; •■ ■. : t. ' ttolir many variations may be made of the letters in the word Sacchanalu ? * iXaf^number of rs^= 2 - 1X2X3 =4(=cnumber of tfs)=r24 I K 2 X a X4 X 5 X 6 X fX 8 Xp-X IQX *A(;j3pumbcr of kt- tersUiuie'woTd)=:399i68oo - , \ J'. . 2X24=48)39916800(831600 the ansWtn '^-^ ■■ ■ ici' ■• * ■■•' *'••■ ' ;• 76 , . ; . ^ HcMn one ; which >is sz ^ ^3^^ ;:;:: i. Again, if three of the qoan- 1X2x3X4 lallfes only be' the same, or^ki become 4faa{,:^:iiumbcr«'of va. nations will be reduced to these 4, aaaif .0aia% /^aa andteuri which Is = 'XaX3X4 -i 4^ j^nd diusit inaV be shewn, that, 1X2X3 .■ . . J .-.'. .-.-.. ,:.■ » if two of the qaandtiesfbe alike, or the 4 .^^tities be aaicf the number of variations will. be. reduced to 12 ;■ which my be* tx^^ pressed by I2i!2l*iif-=:' 12. And by reasoning in the same manner, it will appear, that the nomber of changes, which can be made of the.quantiti^,<i^^r, is equal to 60 ; which may be expressed by ''X^>^3X4X5X / IX«X*X2XS rs6o ; and so of wy other quantities whatever* ajO A&ITHIf BTIC« 0. How many diflFcrent numbers can be made of the following figures, i:;52ooo5555 ? Ans. i:^6p^ 3. What- id the variety in the succession of the follo#^ ing musical notes, fa, fa, fa, sol, sol, la, mi, fa ? Ans* 3j6o. PROBLEM I^« To find the changes of an) given ffumber of things^ taken a given number at a time ; in which there are several given things of one sort^ several of another^ istc. , RULE.* 1. Find all the diflferent forms of coihblriatidn' ojf ill iLt given things, taken as many at a tim>; as in the question/ 2. Find the number of changes in any form, and mul- tiply it by the number of combinations in that form. 3. Do the same for every distinct form ; and the sum of all the products will give the-wkele number of changes required. Note. To find the different forms ^ combination proceed thus : ' . \ . ' ■ I*. Place the' things so, thiat the greatest iri^ces 'may b< first, and the rest in order. 2. Begin with the first letter, and join it to the ^cond, third, fourth, &c. to the last. 3. Then take the second fetter, and join it tb-thd tWircl^ fourth, &c. to the last ; and so on- through the wfcdlei always remembering to reject such combinations as have occurred before ; and this will give the combinations of all the twos* 4. Join * The reason of this rule is plain from what has been shewil before, and the nature of the problem. PBRMUTATION AND COMBINATION. ft} I 4. Join the first letter to every one of the twos follow- ing it, and the second, third, &c. as before ; and it will give the combinations of all the threes. ^. Proceed in the same manner to get the combinations of all the fours, &c. and you will at last get all the sev- eral forms of combination^ and the number in each form* EXAMPLES. I. How many changes may be made of every 4 letters^ that can be taken out of these 6, aaabbc ? No. of Forms. No. of com- No. of changes in forms. binatioDs. each form* • rix*xjX4=H 1st fl'^, a^c z < — =4.. (.1X2X3 =^ ("1x2x3X4=24 tiXiXiX2=:4 1X3X4=24 3d a^Scf b^ac i -c — ai^k* 0x2: liX.z ss z . 4X2= 8 T.:^X:;%?=:.24 .33 == the number of changes required^ tr. How many changes can be made of every 8 letters out of these 10, aaaabbccde i ' ' Ans. 7j2^6q. ': 2n How rnany diflTerent n^^mbcrs can be made out of i iwit, 2 twQS) 3 threqs, 4 fours, ^nfi 5 fives^ t^fen- 5 at;* SJ) ARITHMETIC* PROBLEM V. To find the tiumher of combinations if ofty given number of tbingSf all different from one amther^ taken any given num» ber at a time. RULE.* t. Take the series i, 2, 3, 4, &c. up to the number to be taken at a time, and find the product of all' the terms. a. Take • This role, expressed algebraically, is Jl X -2lll x ^!^^ X .12 3 ? ^. , &c. to n terms ; where m is the number of given quanti- 4 . ties, and II those to be taken at a tim^ DEMONSTRATION OF THE RuLE. I • Let the Dumbef of thiiigi to l^ taken at a time be 2, and the things to be combined zznu Now, when w, or the number of things to be combined, is on- ly two, as a and by it is evident, that there can be only one com- bination, as edf ; but if /ra be increased by i,' or the letters to be combined be 3, ^s abc^ then it is plain, that the number of com" binations will be increased by 2, since wi£h each of the former letters, a and 3, the new letter c may be joined. It is evident, therefore, that the whole number of combinations, in this case^ will be truly expressed by 1+2. Again, if m be increased by one letter more, or the whole number of letters be four, as abed j then it will appear, that the whole number of combinations must be increased by 3, since with each of the preceding letters the new letter d may be com- bined. The combinations, therefore, in this case, will be truly expressed by 1 + 2-1-3, Ia vB&MtrriTioii Am> 6oubimatiom. 233 a. Take a ecrics of.n msaxf terms, decreasing by i, from the given number, out of which the election it to be made, and find Ac product tt tSLikt tcrms^ ' 3. Di^de the last product by the former, and thic qtuit ticnt win be the rtumber'songfat* - ■ a - , . - In the same manaer, it niay be shewn, that the whole number of combinations of 2, in 5 things, ^hc 14-1-1-34-4 ; of 2, in6thbgs, 1+1+3+4+5; and of 2, m 7, 1+2+3+4+5 + 6, &C. Whence, mtlrcrsally, the mmxber of combinations of m thingSf fttkta a by t, it nk+t+3+f+f +iS, ifc;^ id t^i tMdsL * i * . ' * . ! . -.'-.' . But tht som of this scries is z^+ fLUa ; which is the same at I a the rule. * .- 2. Let now the onmbcr 9( qaaotitiQi an. each cotubinitioa be t^yposed to be three. ^ '■ ■ ■ { • ' Then it is plain, that wb^ m^j, or the things to be ^iomlMit are ale, there cai^ be only ode consbitatiOD $ but if in be tncrfaMi by I, or the things to be combined be 4, as akd^ thep wiU.'dit number of combinations be increased by 3 ; since 3 is the num- ber of combinations of 2 in all the preceding leUers aic, and with each two of these the new letter d may be combined. The number of combinations, ^erefbre, in thts case, Is 1+3. Again, if m be increased by one more, or the number of let- ters be supposed 5 ; then die former number of combimctions^yill be increased by 6 $ that is, by all the combinations of 2 in the 4 t)rccediog letters, abed ; since, as before, with each two of these the new letter i may be comtnisred . * The nnmber of combinations, therefore, lii (lus case, is t/' +3+6. ^ r' Whence, unirersally, the number of combinations of m things^ taken 3 by 3, is 1 4.34.6+10, &c, to m— « terms. X. How nunytXioqigyiHifw C9HM^JBadei.<% .6 kusf ,l»X 9M(BX7X6X5(=: same ntunber from 1 0}= 1 5 1 zoo 720)151200(210 the aiuver. ~- mo'' """ • '- -^ , ■ ■ ■■^*+ out of 24 letters of tbc alphabet I 3. A general, wEo KaJ often been successful lli'lmi Wat asked by U^Kfaig^ whit jeWaird ke shoirid* <^fii^^ ii]^n him for his senrices i the general only db^Arcd' a^'fiiMSfllgf fpr every file, of lo men in a ill^^^'i'^^ 'he>e6tid'fliake witb a body of ioo< men : what ir the* imount inrfbtttids •crrling ? - '*' - ^* Aw<iK)3*y7a35t>L 9s. 2dS . But the sam of this scries is = iH xJ^H? X 'J^ 5- which fe ' I . 2f ; 3" tht same as the rule. And the same thing will holdf^ let tho imitiier of dniigs».to be taken at a time, be what it may ; therefoife the number o£ com- Ifaatidnsofm things, taken f»at a^^ne, will s ^X^^^ — ^ X^^^ X. -' 1 - a S m2LJ, 8cc. to» terms. Q^E. D. ^ 4 FERMUTATZOig A;iliiD COHBIKATIOM. ^5 r PROBLEM VI. T^ojind the number of combinations of any given number of things y by taking any given number at tt time ; in nvhicb *^ '.there are sevfratibmgf of^e sart^ severid ef im^Cher^ ytJ l^t7L£» 1. Find, by trial, the numlicr of different forms, wftich th^ things, to be taken at a time^ will admit of, ^nd the num* bcr of. combinations m each. 2. Add together all the combinafions, thus found, and ^ sum will be the number required. EXAii?LES. |. Let the tlun^s proposed be aaabbc ; it is required to £nd the number of combinations t^at can be made of every three of these quantities. - Forms. ComUnatkmB. a^ I fx^b^ flV> i% b*c 4 , .'# -. a& ' I - • ^"""^ <6 T=; number of combina* dons required. - : ' a. Let aaahbbce be. proposed j it is required to find the niimber of combinalions of these quantities^ taken 4 at a Ans. 10. 3. How many combinations are there in aaadbbccde^ 8 being taken at a time ? Ans. 13. "4; How many combinations are there in aaaaahhhbbccccdd ddeeiee^gy to beiug taken at a time ? . ; PROBLEM i^4 iamuMTxtu P&OSLEM VII» Xe-^ndtbi comfukknt ^ amy m^aJker^ tHM iftfo/ numkr^ sets^ the things themselves being all different* RULE.* Multiply the number of tlungs in erery set condnuall^ together^ and the product will be the answer required* EXlMPLSHf. * Demovstratiow. Sappose there are opiy two sets \ tjbe% it is pkin» that every quantity of one set» being combined widk every quantity of the other, will make all the composdioQt oC two things, in these two sets ; and the number of these coii^fott- tions is evidently the product of the number of quantities in w^ set by that in the other. ^ Again, sappose there are three sets ; then the compoikitlba oC two, in any two of the sets, being combined with every qoantity of the third, will make all the compositions of 3 in the 3 teta^ That is, the compositions of 2, in any two of the sets, being moL ' pplied by the numbcf of '^ quantities in the remainiag set, w31 pro. duce the compositions of 3 in the 3 sets ; whtch^^ efidsntiy the continual ptodua of all the 3 nu(nbers in the 3 sets. Ail4 the same manner of rea^ning will hold, let the number of sets be yrhatitwiU. (^E. D. The doctriqe of permutations, CQmbinationSi -ftic. is of vei^jUp^ tensive use in different parts of the mathematics \ particularly il) the calculation of annuities and chances. The subject might \ifB(% been pursued to a much greater length ; but what has been dkis ^eady will be found sufficient for most of the purposes to ^hid| ^tisgs of t^is nature are applicable. "' ■ . 1 i. Suppose there «ic 4 eompaniea, in fSick of wldch tberc; aie 9 men ; it is required to find Boi^r many ways 4 jMSi^ piay be^cho^eny ene wt^f eaph company ? . «56i ' Or, 9X9X9X92:6551 ttie.answer. t. S gpp i Mie "fahcra. WB 4 companiesi in one of which ihereHunf'^ inkh, in anoAer 8, and in each of the other two 9 $ ifAat are die choicesi by a composition of 4 men. Ode "out of eadh company ? Ana. 3888. 3; HoMT many changea are there in throwing 5 dice ? y - . Ana. 7776. I mmmmmmmmtfKmmii «. Wh Alf difietence ia there betwew twice fifc and twenty, and'twieetwcnty^te t Asi. acM \ %,' A wajs |x>rn when B waa at yeara of age ;. how old wip AhewhenBia47i aiid liirhat wiD be the a{;e oJ^^ when A is 60? A^Aad^Bth -^ 3. What 3.. Whtt numheXf tsktti tutm the iqturo of 48, ml leave 16 tiinet 54.? Ans. 14^- 4. Wiiai xnTrnKri aildcd to. dbe thit^fixtt pitft of 3^13^ Wall make ^som 200 ?' " *"' ; Ans. 77; 5. The remainder of a drHmn is* 3259* die'qaodeiit 467, and the diyisor is 43 more than the sum of both < what is the dividend ? Ans. 390270* 6. Two persons depart from the same place at the same dme } the one travels 30, the other '35 miles a da^ : how £ur are they distant at the end of 7 dajr^' if thejr travel both the same road i and how fuTj if diey travel in contrary di- rections ? Ans. 35, and 455 miles- 7. A tradesman increased his estate annually by zooL more t^an -^ part of it, and at the end of 4 years fomidy that his estate amounted tQ 10342!.' 3s. 5^ What had hs at first ? ' Ans. 4000L S. .Divide 1200 acres^'laadan^qiBg Af:]^a3y|<r(^«>.^t r B may It^ve .100 more than A» aad:C44 mqjKf^'tfiiui^lil^^, -^ • Ans.A3ia,A,4%ajjdd^ 9, Divide 1000 crowns i give A lao more, imq.^ j^ less, than C. ■ . . Ans. A 445, B 230, C 325. lo* What sum of money will amount to 1321* l6s.' 3d. in 15 months, at 5 per cent, per annttm^ umpk interest i Ans. 125!^ 1 1. A fadierdivided'Siis Sotttmt among^liis seoi, giving A 4 as often as B 3, and C 5 as often as B d ; what was " the whole legacy, supiposing A's share 5000L ? ^ ^ ■ '"*^^'-Ai*ix875L 12. JLiooo men, besieged in a town with provisions for 5 '^V^fVi ^ach man bciiig atlbWcd 'i6oz. a day, were n> infoirc^HKbcSoo men more. On hearingi that they cannot be rdiie^df ST the end *of 8 weeksy how many ou'niocs § day most each man have, that the provision may last tha} time ? Ans. 6jOZ. 13. Wlu^t number is that, to which if ^ of ^' be added| die sum will be i ? Ans. f^ 14. A MZSCELLANIOUS qmUTlONt. 239 14. A father dytng left. his. son a fortuAci -^of vhich he ran through in S monthd ; 4 <>f ^^^ remainder lasted him a tweke-month longer; after which he had only 410L left* What did his father bequeath hii» ? . Ans^ 956L rjst 46^ tj. A guardian paid his wkrd 3500L for 2500I. which he had m bis haads 8 years^ What rate of interest did ho allow him ? Ancl. 5 pei^ cent« • .i6. A person, being asked ihc hour of the day, said, the time . past noon is equal to -^ of the time till midnight* What was the time i Ans. 20 min. past 5. •17. A person, lookiiig onr his watch, waa asked> what was the ttm'e of the day ; he answered, it '» between 4 and 5; but a more particular answer being required; he sfud, that the hout and minute hands were then exactly to* gether. What was the time ? Ans. 21-^ minutes past 4. • 7-1 8i With ra" gallons: of Canary, at 6s. 4d. a g^lon^ I aaixed 18 gallons of white wine, at 4s. lod. a gal. and lof gallons of cider, at 3s. rd. a gaL At what rate must I sell i quart of this eompositionj so as to<' clear 10 per cent ? " Ans. IS. 3"^ tg&. What length must be cut off a board 8^ inches broad, to contain a square foot, or as- much as 12 inches in' length and 12 in breadth ? Ans. r7^im I 20. What difference is there between the interest of 3jt)l. at 4 per cent, for 8 years, and the discount of th« same* sum at the same rate and for the same time ? Ans. 27I. 3^si 21. A father devised -^ of his estate to one of his sons> and ^ of the residue to another, and the surplus to hia relict for life ; the children's legacies were found to be »S7L 3s. 4d. different. What money did he leave for the widow ? Ans. 635L lo^d. 22. What number is that^ from which if you. take y of 1^ and to the remainder add -^ of -^ the sum will be 10 ? Ans. i^ 23- 240 * AUTBVETIC. a3. A man dying left his wife in 'expectation^: iSkA a child would be afterward added to the surviving family } and making his will ordered^ that, if the (ftild were a son, •|- of his estate should bebng to him, and the remainder to his mother } but if it were a daughter, he appointed tho mother y, and the child the remainder* But it happened, that.tfae addition was both a son and a daughter, by which the widow lost in equity 2400I. more th^in if there had been, only a girl : what would have been her dowry, had shtf had only a son ? Ans. 2ipoL 24. A young hax^ starts 40 yards before a .grey*hoinid, and is not perceived by him till she has been up 40 sec* ends ; she scuds away at the rate of 10 miles an hour, and the dog, on view, makes after her at the rate of 18^ How long will the course continue, anfd what will be the ^ength of it from the placci where the dog set out ? Ans4 60^ seconds, and 530 yards rtin^ 2;. A reservoir for water has two cocks to supply ^ } l^y the first alone it may be filled in 40 minuteSi by tlitf ' second in 50 minutes^ and it has a discharging cock, by which it may, when fuU, be emptfed in 25 minufeea. Now, supposing that these fhree cocks aw all left open, that die tisrater comes in, and that the influx and efflux of the wa« ter are always alike, in what time would the cistern be filled ? Ans. 3 hours %o min. 26. A sets out from London for Lincoln at the very same time that B at Lincoln sets forward for London, dis^ tant 100 miles : after 7 hours they met on the road, and it then appeared, that A had ridden i— mile an hour more than B : at what rate an hour did each of them travel ? Ans. A 7^1-, B 6^ milea^ 27. What part of 3d. is a third part of 2d. ? Ans* '^ 28. A has by him i~cwt. of tea, the prime cost of which Was 96I. sterling. Now, granting interest to be at 5 per cent, it is required to find how he must rate it per pound ^\ llISCEtLAMEOtJS QpBSTlQ^S* 24 1 pound to B, SO that by taking his n^otiable note^ payable at 3 months^ he may clear 20 guineas by the bargain ? ^ Am. 14s. ly^' ^terliiig. dip. What annuity is sufficient to pay off 50 millions of |}ounds in 30 yearsj^ at 4 per cl^t. compound interest ? Ansi 289x5051*- 30. There is an island 73 miles in cirqiunference, and 3 footmen all start together to travel the same way about it ; A goes 5 miles a day, B 8 and C 10 : when will they all icome together again ? Ans. 73 days. 3 1. A man, bgpg asked how many sheep he had in hi3 drove, said, if he had as many more, half as many more» and 7 sheep and a half, he should have 20 : how many had he ? Ans. 5* 32. A person left 40s. to 4 poor widows, A, B, C and t) 5 to A he left y, to B -i;, to C y and to D -g-, desiring the whole might be distributed accordingly : what is the proper share of each ? Ans. A^s share 14s. ^d. B*s 165. 6i|d. Cs 8s. 5j|d. t)^s 7s. j^d. 33. A general, disposing of his army into a square^ finds he has 284 ^soldiers over and above ; but increasing each side with oiie soldier, he wants 25 to fill up the square : how many soldiers had he ? Ans. 24000. 34. There is a pri:?e of 212I. 14s. 'jd. to be divided iamong a captain, 4 men, and a boy j the captain is to have a share and a half 5 the men each a share, and the boy j of a share : what ought each person to have ? Ans. The captain 54I. 14s. -yd. each man 36I. ps. 4~d. and the boy 12I. 3s. i-|d. 35. A cistern, containing 60 gallons of water, has 3 un- equal cocks for discharging it j the greatest cock will emp- ty it in one hour, the- second in 2 hours and the third in 3 : in what time will it be empty, if they all run togeth- er ? Ans. 32-j2j- minutes, ^6. In G G 242 ARITHMETIC, 36. In an orchard of fruit treesy 4- of them bear apples^ •^ pcarSj 7 plumbs^ and 50 of them cherries :. how many trees are diere in all ? Ans. 6oo. 37* A can do a piece of wor( alone in 10 days, and B in 13 ; if both be set about it together^ in what time will it be finished ? Ans. 54-f- days. 38. Ay Ef and C are to share loooool. in the proportion of ji -J and y respectively ; but Cs part being lost by his deadly it is required to divide the whole sum properly .be* twten the other two. Ans. A's part is SVM^yrTi and B*s 42857^5^. £NP OF ARJrHMETIC. ■- ^vt' I I ■gr ill M l ■ " ' . n» .i i ^ -^■ 'JtSb^ . I . ' i Wi . 1 I I iii itwfr LOGARITHMS. JupGARITHMS arc numbers, so contrived, and adapm ed to other numbers, that the sums and differences of the former shall correspond to, and shew, the products and quotients of the latter- Or, logarithms are the numiericaj exponents of ratios. ; or a series of numbers in arithmetical progression, an- swering to another series pf numbers in geometrical progression. Thus Qr Or It h 2, 3, 4, 5, 6, indices, or logarithms. 2, 4, 8, 1 6, 32, 64, geometric progression. I, 2, 3, 4, 5, 6, indices, or logar. 3i 9i ^7* 81, 243, 729, geometric progress. I, 2, J, 4, 5, md. orbg* 10, 100, 1 000, 1.0.000, 10Q009, geom, prog. Where it is evident, that the same indices serv^e equ^illy for any geometric series ; and consequently there inay be an endless variety of systems of logarithms to tlie same <jommon numbers, by only changing the second term 2, 3, or 244 LOGARITHMS. or lOy Sec* of the geometrical series of whole numbers | and by interpolation the whole system of numbers may be made to enter the geometric aeries^ and receive their pro* portional logarithms^ whether integers or decimals. It is also apparent from the nature of these series, that if any two indices be added together^ their sum will be the index of that number, which is -equal to the product of the two terms in the geometric progressipn, to which those indices belong. Thus, the indices 2 and 3, being added together, make 5 ; and the numbers ^ and 8, or the . terms corresponding to those indices, being multiplied to- gether, make 32, which is the number answering to the index 5. In like manner, if any one index be subtracted from an- other, the difference will be the index of that number^ which is equal to the quotient of the two terms, to which ^hose indices belong. Thus, the index 6 minus the index 4=2 ; and the terms corresponding tQ those indict are 64 and 16, \yhose quotient zr4 j which is the number an^ swering to the index -2. For the sanr^e reason, if the logarithn; of any number be multiplied by the index of its power, the product will be equal to the logarithm of that power. Thus, the index or logarithm of 4, in the above series, is 2 ; and if this number be multiplied by 3, the product will be Z^6 ^ which is the logarithm of 64, or the third power of 4. And if the logarithm of any number be divided by the index of its root, the quotient will be equal to the logar- ithm of that root. Thus, the index or logarithm of 64 i^ 6 •, and if this number be divided by 2, the quotient will be 11:3 ; which is the Ic-jarithnri of 8, or the sauar^ root cf 64. The logarithms most convenient for practice are such, .IS are adapted to a geometrical series, increasing in a ten^? fold proportion, as in the last of the above forms 9 and are MATURE OF LO€^AR^THMS* £45 those, which are to be found at presenti in most of the common tables of logarithms. The distinguishing mark of this system of logarithms iS; that the index or logarithm of 10 is i ; that of 100 is 2 ; that of 1000 is 3, &c. And, in decimals, the logar- ithm of 'i^ is —I 5 that of 'oi is —2 5 that of 'ooi is 1—3, &c« the logarithm of i being o in every system. Whence it follows, that the logarithm of any number between i and 10 must be o and some fractional parts ; and that of a number between 10 and 100, i and some * fractional parts j and so on, for any other number what- ever. And since the integral part of a logarithm, thus readily found, shews the highest place of the corresponding num- ber, it is called the indexy or characteristicy and is common- ly omitted in the tables ; being left to be supplied by th^ person, who uses them, as occasion requires. Another definition of logarithms is, that the logarithm of any number is the index of that power of sbme other number, which is equal to the given number. So if there be Nzzr", then n is the log. of N^ where n may be either positive or negative, or nothing, and the root r any num- ber whatever, according to the different systems of log- fi^thms. When n is no, then iV is izri, whatever the value of r is ; which shews, that the logarithm of i is always o, in every system of logarithms. When « is =1, then ^is zzr ; so that the radix r is fdways that number, whose logarithm is i in every system. When the radix r is =2'7i828i828459, &c. the in- dices n are the hyperbolic or Napier's logarithm of the numbers N'^ so that n is always the hyperbolic logaritlim n of the number iVor 2*718, &c.| . But when the radix r is =10, then the index n becomes jhe common or Briggs' logarithm of the number N ; so that %^6 tOGAUTHM^ that the common logarithm of any number xo* or iV is «. the index of that power of lo, which is equal to the said number. Thus, loo, being the second power of io» will have 2 for its logarithm ; and looo, being the third power of 105 will haye 3 for i^ts logarithm : hence also, if 50 be --^Qi-tfp«9 7^ then is 1*69897 the common logarithm of 50. And, in general, the following decuple series of terms, viz. xo*, 10', lo*, 10*, 10°, 10""', fo"*, xo""S io""S or lOOOO, 1000, 100, 10, I, •!, 'ox, 'GOI, 'ooox, have 4, 3, 2, I, o, — i, •—2, — 3, —4, for their logarithms, Respectively. And from this scale of numbers and logarithm^ the same properties easily folloW| as before mentioned. PROBLEM. 21? compute the logarithm to any of the natural numbers^ l| 2, 3> 4» S> ^^• RULE. Let b be the number, whose logarithm is required to be found \ and a the number next less than ^, so that £— -azz I, the logarithm of a being known ; and let s denote the sum of the two numbers a-^-b. Then 1. Divide the constant decimal '8685889638, &c. by /, and reserve the quotient ; divide the reserved quotient by the square of /, and reserve this quotient ; divide this last quotient also by the square of j, and again reserve the que- rent ; ^nd thus proceed, continually dividing the last quotient by the square of /, as long as division can be made. 2. Then write these quotients orderly under one anoth- er, the first uppermost, and divide them respectively by the COMPUTATION OF LOGJARITHMS. 247 the odd numbers, i, 3, 5, 7, 9, &c. as Ibng as dmston can be made ; that is, divide the first reserved quotient by i, the second by 3^ the third by 5, the fourth by 7, and so on. 3. Add all thes6 last quotients togetheri and the sum vrill be the logarithm o£ i-^a y therefore to this logarithm add also the given logarithm of the said next less number Of so will the last sum be the logarithm of the number i proposed. That is, log. of i is log. j'+ - X -f &cl where n •S685889638, &c. denotes the constant given decimal EXAMPLES. ^ Example i. Let it be required to find the logaritlim of the number 2. Here the given number b is 2, and the next less numter « is I, whose logarithm is o ; also the sum 2-f-i=3rD-, and its sqikare s^zzg. Then the operation will be as foU low« : 3)*868588964 i)'289S296s4(-289S29654 9)-2895J9654 3) 32i69962( IQ723321 9) 32169962 5) 357444o( 714888 9) 3574440 7) 397i6o( 56737 9) 397160 9) 44n9( 4903 9) 44129 ") 4903{ 446 9) 4903 13) 545( 4^ 9) 545 15) 61C 4 0) 61 Log. of -f Add log. I •30102999; 000000000 Log. of 2 '301029995 Example 248 ' LOGAUlTHItS** Example % Td compute the logarithm of the humter j* Here tz^^* *^ ^^*^ ^^^^ number a=2, and the sum u+^i=5ri:/, whose sqUare /* is t^ to divide by which, always multiply by '04. Then the operation is as follows i 5)*868588964 25)*>737i7793 25) 6948712 25) 277948 25) 11118 45) 445 18 i)-i737i7793(-i737i7793 3) 69487 I 2( 2316237 5) 277948( 55590 7) iin8{ 1588 9) 44S( 5° n) 18( 2 Log. of -f •176091260 liOg. of 2 add •301029995 Log. of 3 sought '477121255 Hieti, because the sum of the iogarithn|S of numbers gives the logarithm of their product, and the diSerence of the logarithms gives the logarithm of the quotient of the numbers, from the above two logarithms, and the logai^ ithm of 10, which ^s i, we may raise a great many log« arithms, as in the following €jxamples : ExAMPtfe 3. Because 2'X2=:4> thereforeJ To logarithm 2 "3010299951- Add logarithm 2 •3010299951- ■■■■■> ■!■ Sum is logarithm 4*60 2059991-5-. EXAMPL£ 4. Because 2X3=6, therefore To logarithm 2 '301029995 Add logarithm 3 •4771^1255 Sum is logarithm 6*778 15 1250. EXAMI'LE QQUBxrcM^rmuw jjoqsmituvm. tap Because a^ s: 8» therefore Logarithm 2* *30iP2999jr'r Multiplied Ir^ 3 ^ Girea Logaiitfam' 8 ' ^5^3039987^ Because 3* rr'^p, therefore Logarithm 3 •477121254-1^ i a Gives bgarithm 9 '954242509. . :£xAMPLfi 7. Because ^ zr 5, therefore From logsrithm; 10 i '000000000 Take logarithm 2 '301029995^ Remains Iqgarithm 5 '45989700041* Example 8. Because 3 X 4 ::^; 12, therefore To logarithm 3 •4771 21255 Add logarithjp 4 1502959991 Sum is logaritlim 12 .i;p 79 181 246.. And thusj computing hj this genend nile> ihe iogxr^ ithms to the other prime numbers ^f 1I9 I3» X7> iJH 23, &c« and then using composition and d|vi$ion| vrt may ea-» , . siljr •ily find as many logarithmf as 4tre glease^ or may speedily examine any b^arkhm in the table.* Dsscjujnriojfr akd Usr^qh t&b .TABLE of LOGARITHMS. Integral nitmbers are supposed to form a geometrical se« ties, increasing from unity toward tlic left ; but decimals are supposed to form.^,Uke. series^ decreasing from unity toward the right, and the indices of -their logarithms are negative. Thus, -f-i is the logarithm of ro, but -r-x is the logarithm of -j^, or 'i ^.and -f-a is tbe logarithm of 100, but — 2 is the logarithm of ~^ or 'ai ; Mad so om Hence it appears m general, that aD numbers, which consist of the same figures, whither they be integral, or fractional, or mixed, will havie the decimal parts of their logarithms the same, difieriti^ only in the index, which will be more or less, and posidreior h^tSve, according to» the place of the first figure of tie iramber. Thus, the logarithm of 2651 being 3*4234097, the logarithm of —^ or -—-^y or tToT> *^" part.of it, will be as follows : Logarithms. 3*4.^34097 2-4234097 i*4^34097 0-4234097 — r4234097 ■•-i^2-4234097 —3*4234097 Hence f Many other Ingenious methods of findiBg the logarithms of numbers, and peculiar artifices for constructing an entire table of them, may be seen in Dr. Hutton's Introduction to his Tabki, ;md Baron Maserjbs* Scriptores Logarithmlcu * DE8CR1PTIOK .010 «n MlTHB TABU. t^l . Hqidc U'appeatf, tbxt the'iiidez^ occharurterUtk, «f atnf iofarillmi itadwaysJessby r fhabthriiumber of-inte& fpa.Bgkr^ wlnckifaf^iiatiiM.nuabcribcmiists of ; cnritit mpai to^tlii&difltuice of thJb first or left-lund iBgture from the place. of nhilv^'oq fynlipiabeofiiiitegecly whether <Ni tlie lefty or.oa tfae-sigfat of it & Md^tfairindex is constant- ly to be placed ttO' lhe left of die deouul part of tUe logaritl|iBu i . When there fre ixitegers m the gtjraiwAac, the index is ^ways af&rmatiFe ; but wKqd th<ere.Me*iia;im«gei:s, jdbc index is negatir«^ and is to be marlped bf a short line Arawr» before, orf^bove, it. Thu|,|k-J^ttmber having I, ?i 3» 4i 5> &c. int(g|;er places, the indei^ of, its logarithm if ^> h 2, 3, 4, 8fc. ox I le«8 than tie number of those placet* And a decimal fracdon, having its first: figtfre in the ist, 2d, 3d, 4th, &c. plaice of decimals, has always ^-h^i, —2, ^"•3* "~"4» &c, for, the index of its logarithm* It may also be" observed, that though the in£ces of . fractional quantities be negative, yet the decimal parts of their logarithms are always aj^rn^tive'. h To HMO9 IK T0S Tabxs, the Logaetthm to aky I. Iftyniiinher do^ not exceed 1 00000, the decimal part of the logarithm is fo4nd, by inspection in the tablet standing against the given number, ~ in this manner, viz. in most tables, the first four figures of (He given number are * The Tables, considered as the best, are those of GARmNsa ^ 4to. first published ia the year 1744 f of Dr. H^ttok, in ,8vo. first priated ia .1785 i of Taylox, in large 410. published ia 1792 ; and in Fraacei those of CAtLcr, the ^ecood uUtion laUishedio 1795. ^ aft fve iar die §tU coknm of die page» and-tlK fifth fignie m the vppennott Uae of it y then in ^.mglt of. meetii^ $xe the.laic fear figvm oftfae logiUlhMiy mod the fin| three figwm'of die same^at dib fcegiiiriiy otike mvpm line I iDiAichif totbepaefix^jdKpiDpesiiideK; «- If yii ^, So the logftiidn ol .^*ogiz U ^*f 3^5^f that it, d» McmA S2z6$2i^toimdintim taUe^ irith:die index i pre* fixedy becanae t&e gireii number contains two integers. 2" But ^Ihighm tmmhr epfttam mm ibanfioe figuru^ take oift the Ibi^andim 'of the first fii^e figures- by inspect tion in the tilble as before, as aj^o die*nett greater log« aridim, 'subtracting one logaridim from die odier^ and' af- SD- one pf their Itorresponding ntuhben from the' odien Then say, - '. As the difierence between the two numbers Is to the 4ifierence of their logaridunSf So is the rexnaining part of ^e given number - To the proportional part of the Ipgarithm. Which part being added to the less logarithm, befoiie taken out, the wholj: logarithm sought is ;^tained very nearly. EXAMPLE. To find the logarithm of the number ZKO^i^f^, The log. of 3409200, as before, is 5326525, and log. of 3409300 is 53^6652, the di$ 100 and 127 Then, as 100 : 127 :: 64 : 81, the proportional part* This added to 5326525, the first logarithm. gives, with the indcif, 1*5326606 foy the logarith^^ 9? 34-09264. Or, in the liest fables, the proportional part may often be taken out by inspection, by ,means of the small tables pf proportionsd parts, placed in the margin. ' .■ ■ " - If DEsctiPTioK Arm nstt of the tablu^ 233 . If tbcfimiAer ocmsist both of integers and fmctiant, 4oir be emmlj fractional, find ti^ decbnal part of the logan- ithm, as if all its figure^ were inlegfal ; then this, the proper cfaai^oteristio being prefixed, will glv« the logaridim lequired. ^ ^ • - "^ And if the given iluthber be a proper fraction, subtract the logarithm of the dehdininator from dbe Ibgarithm of the numerator, and the remainder will be the logarithm sought ; which, being that of a decimal fraction, must al- wayli have 9 negative index* ■ But if it be a mixed number, reduce It to ah improper fraction, and find the diflference of the logarithms of the numerator and denomixutof, in the same manner as before- I. To find the logarithm of |^ - Ijogarithm of 37 1*5682017 . Logarithm of 94 1*9731279 Diff. r<5g. of |-;- — 1*5550738 Where the index i Is negative. 2. To find the U^arithm of x74-f^ First, l^^ = *^N T^hen, Logarithm of 405 2*6074550 Logarithm of 23 i*36i7278 Plff. log. of 17^1 1-2457272 II. To riNp, IN THE Table, thr, natural NuMBEit to ANT Logarithm. This is to be found by the reverse method to the former, namely, by searching, for the proposed logarithm among those in the table, and taking out the corresponding num- ber ber. hf impectioiif b wliich the jj^per amnber jo£ btegeit it to be fointed ofi^ viz* i more tbui the iiaits of tbe aC* Jfexqitme iiidiBx< I J^i in finding cfe nwtiber aB8W!erifq<to ^ Mijr ^fcn Idginliimi the lodeK vhrnfr ahewt bo# £ur the first figure must be removed from the place of liatta^ to 44>elftft 9f:'^i'Bbcget$, whea die indesE U'affirttathfie;ii..bat to the rijgjhfc or in dtcimaltf whoi it is aegatire^-. BXAHFLES. Soj the number to the logarithm* 1*5326525 is .34*0;^ .And the number of the bgarithm •--*i'53a65a5 is *34092« But if the l^arithm eanfid h exaetij found tH thi tatk, take out the next greater and die next less, subtracting one of these logarithm! from the other^ and also one of their natural numbers from the other, and die less logar« ithm from the logarithm proposed* Tlien sa 7, . As the first diffcrencei or that of the tabular logtfithmsy Is to the diff:l»ncc «of their natund numbers. So is the difierence of the given logarithm and the last tabular logarithm To their corresponding numeral difierence. , Which being annexed to the least natural number above taken, the natural number corresponding to the proposed logarithm is obtained. EXAMPLE. Ilnd die natural number answering to the given logax^ ithm i'53266o6. Here the next greater and next less tabular logarithms^ with their corresponding numbers^ &c. are as below .: Next greater 5^26652 its num. 3409300 ; giv. log. 5326606 Next less 5 3 265 25 its nun^ 3409200 ; next less 5 3 265 25 mmmmmaim^t^immmmm ■■■l^Ba^^ai^Ha^iM w^mm^mmmi^^mmm Di ff e r ence s 127 100 81 Then, MULTlPUCAi:iPH % LC^AftXTHMS. %gp # Then, as 127 : 100 :: Ai ; 64 nearly, the numeral dlflerence. Therefore 34*09264 is the number apnghti two integers being marked off, because iiie index of the given logarithm is I. Had the index been negative, thus, — r53266o6, its corresponding number . would have been '3409264, wholly decimal. » - ; -. Or, the proportional numeral difference may be found, in the best tables, by inspics^tion of the 'smsdl tables of fso* portional parts, placed in the margia« MuLriPuiijrjoN bt Logajlitbms. RtTLE* Take out the logarithtns of the factors firom the tabic, then add them together, and their sum wQl be the logar- ithm of the product required. Then, by means of the ta* ble, take out the natuiral number answering to the sum, for the prodjict sought. * . . - . Note i. In every operation, what is carried from the decimal part of a logarithm to i|s index is affirmative \ a^d is therefore to be added to the index, when it is affirma- tive | hut subtigcted, when it is negative, v? .... . i NoT^ 2. When the indices have like ^{gnsy that is, both 4* 01^ both --*^ they are s to be addc^ and the sum has the common sign ;. but when they, hiiTt: unlike signs, that is, one%'-|- and the other -— > tl^^k 'ilifference, with the sign of the greater, is tp be taken lor ^ index of the (um. -.. . fiXlUftSS. Ijtf COCA&fTHMt; , I. To multiply 23*14 by 5*062. Nttinbers. t^pxhhxn^' 23*14 '^^3643634 5*062 6*7043221. Product n 7*1347 2*0686855 . a^ To xAiiktply a"58i9i6 by 3*457^9^* Numbetsi I^ogarxthms^ 2*581926 0*4119438 3*45729i ^-5387359 .^'->i Product 8*92647 0-9506797 3. To multiply 3*9021 ^97*16 and *03 147318 all to^ gether. . . Numbers. Logarithms* 3-902 0-5912873 597-16 2-7760907 *03i472S — 2*49j^9353 Product 73*33533 1-8653133 Here the —2 cancels the 2, and the i, to be carried from the dechnals, i^ set doMrh. 4. To multiply 3*586, 2*1046, 0*8372 and 0-0294 all together. • ■ Numbers* liogarithnis. 3-586 0-5546103 fio46 0*323 X696 • 0^8^72 — 1-9228292 9- 0*0294 —2-4683473 Product 0' 185 76 1 8 •—1-2689564 Here ' Here the 2, to be came4|^ ^aoo^U t^^ tt4liietstr&. mains the — i to be 4^et down.' Jkimmr BT Lo0dMtiiMiK :- ^ :; '. .- • 'rot ;•■ •• '- . r :j: .•" N. •:. ■ .; !. ■ ' . c .'. ^-•- ." i' .. : ;.-^T ;^/»i.3 t«i'-M'»' .;-- .-in\.*j'Hi From the logarithm of the dividend subtract 'Iftii'tt^ft ^ ithm of the divisor^ and,tlie.9wb^g^wa$bgjt%'^^ faaitider will be tbf fuotient requirei^]'.:/ Note. If -i fcc to !)i5*n:arried to lifeinicf of ||c sub- trahend, applf it i&drAn^ to the ^nf Itf thc'itiiKrS then change the sign of the Index to ^, if it be 4^-i,W\to +» * if it be — -} ai^ proceed accoi^dld^t^ t&ib second ^twte un«. '-; .. ••:. -r.n der the last rule* ^ . ,* EXAMPLES. f . To divide 24163 hf4S&pr Num. Log. Divi(ie«le.ai»i^»^ * ^^^ V4^H3X5^^ \ Divisof 4567 3'65963i<> Quotient 5*2907*^ *'* 0'7a35i99 •: -'.'••. . : . ^ ..:* '1.' frmprrrr': 'Ar'ruTJa , ^..IVdiVldc 77i49'bf 523'!7»'^ ':^'^ ■^''- \ • - '• '-'^ «* y Num. Log. :I8Tidc«d;fl|7J«49^: * ,-.l'i4$949l>. '. .:.roM : JQiviccir. 5aj:7$ XJl^^l^i* ■" ,;;. ;^::.fl[ .^•.,'n J. Divkk -0^314 by *oo734f. Num. Logr Dmcfend '063 14 *— 2*8003046 Dmsor. '007241 •*"3*8S97985 QuotiQiKt S^lij^a o*9405o6r ' Here i, carried from tlie decimals to the —3, makes k bcccmie*— »2^ whichj taken from the other -<»2| kares • 4- To divide 74^ iqr 12*9476. Num. ^ log. Drridend 7438 •— i-87i456a JDhisor 12*9476 1*^121893 Qluotient '05744694 —2.7592669 . Here the ij taken frctm the «— 1» makes k become *•*-% Ip be set down. iNrOLUriON MT LoOJRtTBJf^k HVLJU Multiply the logarithm of the given number by the in* dex of the power^ and the number answering to the prod- act will be the power required. NoTB, A negative index^ multiplied by an affirmative number, gives a negative product $ and as the number carried from the decimal part is affirmative, their cKffinr- ence with the sign of tbe greater iS| in that case^ the in- des of the product iMfoLunm Br LoMftmw. ^ ' BZAHPLSS. • * f# To sqtiare the numl^ 2*579i- Num, • Log; Root 2*5791 0*4114682 ,Thc Index a Power &6s 1 756 o'82ajp364 2t To find the cube of 3*07x46. Num. Log* Root 3-07x46 ' ©•4873449 The Index 3' ' ' ' r . ' ' Power a8'9757y - 1*4620347 J, To raise •091J63 jtp the 4th power. Num. Log. Root *p9i63 *— 2*9620377 The Index 4 Power '0000704938 —5-848x508 Here 4 times the neg^tlTe index being "^89 and 3 to be carried^ the difference -—5 is the index of the product* 4^ To raise 1*0045 *^ *^ 365tb<root. Num. Root 1*0045 The Index mm Log. O'ooip4p9 ' 365 ' 9749S H6994 5849? \ Power 5*i4888t •7"7i35. V , Eroi^riQi^ r lOeAEITHMf^ EroiuriON bt Logjritbms* E0LE. Divide tKe logarithm of the gfiren number by tbe index of the power; and tbe'numbcr answering to the quotient will be the root required. Note. When the index of the logarithm is negative, and cannot be divided by the divisor withotit a remainder, increase the index by a number, that will render it exactly divisible, and carry the units borrowed, as so niany tens^ to the first decimal, place i and divide tlie rest as u&uaU EXAMFLES. I, To find the square root of 365. Num. Log. Power 365 :e)2'5622929 Root 19' 1 0498 V 22 1 146^ a. To find the 3<!ireotxi£ 12345^ y -^ "Niim. Log. Poifirar: 11345 ^^4^09^49*^ Hoot 23"! ii^^ 1*3659364 . ' — 7— . "'< 3. To find the Jtatk root of 5;, au.ai Vm^ togs : Power '2 lo)cx3QXo3oo Root vxrjryjj ©"0301030 . ■ ■ • . Mur .. , > ' — — — 4. To find the: 3$jth root of i'045. Kxmu Log. Power i'^45 - 2^S)^'^^9^^i *^* Root roooiai 0*0000524 5. To * BVOtUtlON BT LOGARlTHMt, atft J. To find the second root of •093, Num. . Ix)g. Power '093 2)-»«2'9684829 Root '304959 ' •— i'48424l5 Here the divisor t is contained exactly once in the ncgf» ative index -^2, wid therefore 'the index of the quotient is —I. / 6. To find the third root of '00048. Num. , ' Log^ Power '00048 3>— 4'68ii4i2 Root -078^^9735 — ;2'^93747i Here the divisor 3 not being exactly contained in •^4t 4 is augmented by 2, to make up 6, in which the divisor * is contained just 2 times ; then the 2, thus borrowed, bc-» ing carried to the decimal figure 6, makes 26, which, di^ Yided by 3, gives 8, &c» £ND OS LOGARITHMS. f ALGEBRA. DEFINITIONS AND NOTATlOR >• Algebra is die art of cfomputing by 8]rmbok* It is sometimes also called ANiLtsxs i and is a gta^ Und of arithmetic^ or uniyersal way of computatioB. 2. In Algebi^^ the givert^ or inown, qutnOitiet are usually denoted by the first letters of the alphaibet, as m^i^e^d^ ftc* and the unirtot(/n, or requited quantities^ by the last lfif« tersy as ^) j^) ^ . Note. The sigfls, or characters, esplained at ^ bo* |faming of Aritbmeticj have tibe same signification in Al* gd>ra. And a point is sometimes usod for X : thtui «+* • fl— -4zrfl-|-iXi»— *. \^ 3. Thqse quantities, before which the sign -j- is placed^ arc called fositive, or affirtHative i; and those, before which the ogn -— is placed, negative. " And it is to be observed, that the sign of a negatiNI quantity is never bmitted, nor the sign^ of an affiVmafive .- 0Q« Sd4 * ILC^RI. , ^ 0|M^ .except »t be a smglc quantity, or thc^first m a.seriet 6i quintities, then the sign -f- is 6rcquentiy*omS!cif ;' iKus n A^pftGes thesame as +<f». ^ ^^ ^ne^ >^'i'i''^HH ^^ tame as -f-'»+*~^+^ 5 ^ ^^^9 if any single quantity, or if ^ the first term in any number of terms, have not a sign before it, then it is always understood to be affirmative. 4. JJh signs are either alL'ppsitivc^ or ^begatbe ; but 4igm are ufiiiie, when some are positive and others negative. {• Single^ or simple^ quantities consist of one term onIy» In multiplying simple quantities, we frequently omit the sign X, and join the letters ; thus, £73 signifies the same as tfX3 ; znd aic. the same as aXbXc. And these proMcfe)' viz. aXK ^ ^> ismd'iMV-ari tUBdS-^Kigk or simple quantities, as well as the factors, vfz. a, 3, r, from which they are produced, and the same is to be observed - <>( the {products, arising fj^om the- mutei{»UcfM;io]^'of any |ll^XlJ)et of sim{)le quantities. \ ' ' ' . . 6. If an algebraical quantity consist rf two terms, it ik called , a binomial ^ as (i^b } if of three terip^ a trinomial^ as a-^b'^-c 5 if of four terms, a qufldrinamufi, .j^ f-^b^ r|-^f and if. there be more terms, it is< c^H^A a mult^^ . nomml^ or potynomiql ; all which are compound quantities* When a compound quantity is to be expressVd* as multi- plied fay SI single one,, then we pbcq th^ sign of jmihipli-- cation between them,^ and dtawft line.ovet! the ix>m|K)iiii4 qusmtity only \ bat wheir compound quantities are . to bf represented as multiplied together, then we draw a Um over each of them, and connect them wifli a proper sigit. ~ Thus, /j-|"^X^ flenptes that the compound quantity ar\^h i4 multiplied by the simple quantity ^r ; sp thiat ijf a vrttt i(^ 1^, and e 4, then would a-^-bXc be io+dX4> ^ i^ i^** to 4, which is 64 5 and fl+*Xr+rf expresses-the product of jDEtlNITlONS AND NOTATION. ^ • 26$ jof tW compound qoantitics oHh* ^^ c+d multiplied to- gether; .; " :,-'.. ,. 7. When we would express, that one quantity, as a^ is greater than another, .as-^, wc write flC"*» or fl>^.j and if we. Would ea^press, that a is less:- than i, we write a'^» .or a'<ik ■'..-!. ' • ■ • ' : ■... . . ; . . 8. Wh«a we would. Cicpress.the difference betw^een two quantities, as a and ^, while it is .unknown which is the greater of the two, we Write them thus> » w^, which de- notes the difference of a and ^. . y^ ...» . ; 9. Powers of the Same quahtities or factdri are the prod- ucts of their multiplication : thus flXa, or 4a, denotes the ^uare^ or second ^ower^ of the quantity represented' by d i aV^aV^a^ or aaa^ expresses the cube^ 6v. third. Jfower ; and flX^X^X^^ or aaaa^ denotes the biquadf-ate^ ox fourth power And it is to be observed, that the quantity a is the root of. all these powers. Suppose aiz^, then will aaZHaXtf^^ 5x5=25== the square of 5 5 flkij=flX^Xa=;sXsX5=; ^a5i=:.the cube of 5.} and aaaai:zaXaXaXaz:;sXsXs X 5=625=1 the fourth power of 5. 16. Powers aire likewise represented by placing above the «>Qt, to the right hand, a figure expressing the number of factors, that produce them* Thus, instead of aa, we write <j* 5 instead oi.aaa, we write a^ ^ instead of aaaa^ II. These figures, which express the number of factors, that produce powers, are called their indicesy or exponents : thus, 2 is the index or exponent of a* ; 3 is that of x^ \ 4 ia that of x^y &c. . But the exponent of the first power, though generally <nnitted, is unity, or i ; thus a signifies the same as ^, namely, K K S^60 • . ..:;.. ai;G£bra. namely, the £i^t {lower of '« ; 0Xit» thp ume afi «! x^, or ii*"f *, that is, fl*, and fl*X« is the same as a^Xa, or j'+'j or <i^ - i' ■ . ' ' ..." .1 • '^ia» In ezpressii^ "powers of compound quantiti^fiy. yirfi uiuiually^dniw a Uhe'xnner the given qu^tnthf, and at^the end of the line place the exponent of the power. . . Thu^ 0-1-^1 d€Mted tht dqnare or second pow^r of a-^^ ' consid* exed as one \quantky ; a+i\ the third* power ( a^i[ thr fourth power, &c. And it may be observed, that the quantity a-\-by called the first power of ^+^, is the root of all these poW^j:§. Let aZZ4 and 6:1^2^ then will a-j^i t)ecome 4+ 2,^ or d '^ and tf+^l ZZ4+2J =:6*=6X6r:36, the square of 6; also a+b\ =4+2J =6^=6X6X6=216, the c«be of S, « . • . ■ . J. ^ 13. The division of algebraie quantities i^ very frequent- ly expressed by writing down the diviser under the divi- dend ^ith a line between themi in the manner of a vulgar a ... fraction : thus, -^ represents the quantity arising by di- viding tf: by r ; so that if « be 144 and c 4, then. will — be --— , or 36. And -^2L. denotes the quantity aris^ c 4 a — c ing by dividing a+i by a— <: ; suppose 5=12, bzz6 and 1 M, «+^ , 12 + 6 18 ^ r=9, then will — — become or — rro. fl— r . 12 — 9 3 14. mcsc iitcrtTi, ci.prct)biuiib, iiamcriy, — aiiu , are called algebraic fracticns ; whereof the upper parts art* call- DEFINITIONS AND NOTATION. 20*7 cd the nuvieratorSf and the lower the denominators : thus, a is the numerator of tRc fraction -r, and c is its denomi« ^ •,'••■■. ■ ~a+b nator i aju^ is the numerator of — r-it.ranrf a— -^^ is 4ts denominator. 15. QuofndtiPSi to which the radical sigah ai^plied^ are called radical gtktntifiei, ^ surds ; trheteof those consisting of ode termottly,. as ^a and ^ a x^ are caHed jimpig sards : add those eoYisisting of several terms, as ^ ab+cd and -^a**— ^*+fc3f compound, surds. 16. When any quantity is to be taken more than once, the number is to be prefixed, which shews how many ^es it is to be taken,; and the number so prefixed is called tl^ numeral coefident ; thus, 2a signifies twice a^ ox a tak- en twice, and the numeral coefficient is 2 ; 3^;* signifies, that the quantity k* is multiplied by 3, and the numeral coeflSicient is 3 5 also S\/a;*+^^ denotes, that the quan- tity ^x*-^a^ is multiplied by 5, or taken 5 timc^. When no number is prefixed, an unit or i is always un- (Kirstood to be the coefficient : thus, i is the coefficient of flor of a; ; fot a signifies the same as I^7, and x the sarflc as ixy since anj' quantity, multiplied by unity, is still the Moreover, if a and d be given quantities, and x* and y leqoiied ones \ then ax'' denotes, that x* is to be taken a times, or as many times as there are units, in a \ and dy Acws,- that y is to be taken d times ; so that the coefficient rf'i2»* is tf, atid that of dy is d : suppose ^n|6 and ^—4, ' ' ' • . then 268 ILCBBllA. then will /T;f*ir&r*i an4 dyz:;:^ ' A^ain^ -*, or — , de-. notes the half of the quantity 96^ and the coefficient of ~a? is \ ; so Ukewiae ^| or — , signified '\ of Xy and the'co-' "4 efficient of -^A? is -I* 17. Zfitf quaftikies are those^ that, are represented by the same letters undf r the ^am^ powers^ .or which differ only , in their coefficients : thus, 3a, 5^ and a are like quanti- ties, and the same is to be undec^tood of the radicals \/x*'i:a* and J\/x*+a* • But td/t/iie quantities ar« those, which are expressed by different letters, or by the same letters undqr different pbwerisJt thus lab^ a*b^ 2abcy sab * , 4;^ * , v> ^ * and z * are all unlike quantities. 18. The doubk or ambiguous sign >^ signifies plus or tni^ nus the quantity, which immediately follows it, and being placed between two quantities, it denotes their sum, or dif- fcrence. Thus, y^+\/ — b shews, that the quan- 4 y^^ tity y/ — - — ^ is to be added to, or subtracted from, ^, 4 19. A general exponent is one, that is denoted by a letter, instead of a figure : thus, the quantity a;'" has a general exponent, viz. w?, which universally denotes the ;»th pow-r cr of the root x. Suppose mz:zZy then will x^^zzx"^ ; if ff| r:r3,.thcn will x^zzx^ ; if mzzi^^ then will x'^zz.x^y &c. In like manner, a— ^j expresses the wth power of a — b^ 20. This root, viz. a — b^ 13 called a residual roct^ be- cause its value is no more than the residue, remainder, or ijiffef^nce, of its; terms a and b. It is likewise call- ed DEFINITIONS Xlfb KOTATION* 26^ \ cd a binomial, as w^ll as aJ^b, because it is composed of two parts, connected together by the sign — . 21. A fraction, which expresses the root of a quantity, i« also csdled ar^ index, or exponent ; the numerator shews the power, and the denominator the root ; thus a* signifies the same as ^ a ; and a+ahl"^ the same as ^a+ab ; likewise a\ denotes the square of the cube root of the quantity a. Suppose 0=164, then will aTzi:64T^zz^^z:: . 16 ; for the cube root of 64 is 4, and the square of 4 is 16. Again, a + b^. expresses the fifth power of the Biquad* Mtic root of a^b. Suppose azig and bzzj, then will tf + ^|"*zr9^+7|^=i6|'^i=2^=32 ; for the biquadratic toot of 16 is 2, and the fifth power of 2 is 32. - Also, d^ signifies the «th root of a. If «zr4, then will fin'^a^i if npzg^ then will flV^fiT'j &c. m ■ MbfebVer^ tf+3|" denotes the /wth power of the «th root rf O'^b. If wz = 3 and «=2, then will n-f-^l" =o-j-3| , liamcly, the cube of the square root of the quantity ar-f"^ > and a^ «* equals y/fl*,or x/a, so ^+^|" = v^HMr» naniely, the «th root of the fwth power of' a-\-b* So that the fwth power of the nth root, or the «th root of the mth 'power, of a quantity are the very same in effect, though differently expressed. ^ 22. "An exponential quantity is a power, whose exponent is a variable quantity, as x^. Suppose ^=2, then will ^=2*;=i:4 } if >=3, then will ;v*= 3^=27. ADDITION. 270 ALCBSRA. ADDITION. AnDiTioy, in Algebra, 16 connecting the ^oantities to^ gether by their proper signs, and uniting in simple terms silch as are similar. In addition there are three cases. CASB I. When like quantities have Rle signs. RULE.* Add the coefficients together, to their sum join the cdounon letters, and prefix the common sign when.neces-^ sarjr. _ £XAMPLBS« ----- ■■ .,-. .. ., _ * The reasons, on which these operations are foUnded^ yil] readily appear from a little reflection on the nature of the quand- tie3 to be added, or collected together. For with regard to the first example, where the quantities are ^a and. 5«,. whatever a represents in one term, it will represent the same thing in the other ; so that 3 times any thing, and 5 times the same things coi- lected together, must needs make 8 times that thing. As, if a^ denote a shilling, then o>a is 3 shillings, and ga is 5 shillings, and their sum is 8 shillings. In like manner >. — 2aB and — 'jaB^ or —2 times any thing and — 7 times the same thing, make —9 times that thing. - As to the second case, in which the quantities are like, but the signs unlike ; the reason of its operation will easily- appear by re- flecting, that addition means only the uniting of. quantities togeth- er by means of the arithmetical operations denoted by their signs -f- and — , or. of addition and subtraction ; which being of con- trary or opposite natmes, one coefficient must be subtracted from the other, to obtain th^ incorporated or united mass. As ,£XAMPl.¥S. • - I. 2. 3. AM r — 7^ 4«3f* 7av— > C^S^ ""^^ "S^J^* &Af— 2y Sum 8a — loi idflfj* /ix— j^ ■HM^MiaM MBM^MM M^MMM MMMMMi^iM^p As to the third case, where tke quantities are unlike, it is plain, that such quantities cannot he united into one, or otherwise atfdecfthan by means of their signs. Thus, for tSMmfkf if « he . supposed to rejnresent^croitrn, «nd b a shilling ; then tlM Aim of a and ^ can be neither ca nor 23, 'diat is, eeitber t crowttft lK>r e shillings, but only i crown plus i shilling, that is, a +5, ^ In this rule, the word^ aJdttion is not very properly used, being much too scanty to express the operation here performed. The business of this operation is to incorporate into One mass, or alge- braic .ei^ess^on, Afferent Algebraic quftodtiee^, aa far 99 aa aciuti incQiporation or* union is possible ; and to. re^in the algebraic inarks for doing it in cases, where an union is not possijble. Whep wc have several quantities, some afErmative and others neg- ative, and the relation of t)kese quantities can hft di^cpvere^ jfi whok or in part ; suck iftcerporatiea of twa <or moM quantities into one is plainly effected by the foregomg rules; It m^y seem a paradox, that what is called addition in algebra should sometimes mean addition, and sometimes subtraction. But the pacradox wholly arises from the scantiness of the name, given tothealgck-aic process, or fron.em|doyiiigan sUteimiB.a'ncv/ wd aiore eAlargdi sense. Instead of addMoD, caU u umrfo^. tkmi union,, or rttP^mg A'lakncij tttd tkc paradox ^WiiAfl.* a72 3*» ATy ' '• j_ * 4^—3* 2Ai— 3«, .. 2^r:-^3*+2^^ 4**— 8^ - a*y-^A:4y8fl^ y^2b W»— 2*y 5*r-3*+ ''* % CASE II. When Hie quantities have ««/»i^ j/^w. - RULE* Subtract the. less coefficient from ti^e greater^ to t!bef> mainder prefix, the sign of the greater^ and aiuiex their common ktteri or quantities. - EXAMPtES. 1. 2. 3-* 4.. • '. S- To +(Sfl —7* +a<: -^cd +3x/«''+** Add— -3a +6* 2f +3^*/ — 8v*'«»»+M Sum -f-4a — i * +2^</ — sv'«'+*' 6. * In example 3, the coefficients of die two quantities, viz. J^te and — 2r> are equal to each other, therefore they destroy one another, and so their sum makes o> or #, which is frequently vsed, in algebra> to signify a vacant place. ADfimfl >K. ^73 & 7- 8, 9- ko. to -f-2« — ^ +7C — 3<i +8v/<»*+4» Add— (Sa +7* ~~^ —tfd -3^^-*+*' +3'' Sum -i-4« +5\/«*+** Note. W^A^/i i«j«y liie qi^antities are to be added togetherp nvhereof some are e^rmative and others negative s reduce them first to twt> terms, by adding all the affirmative quan- tities together, and all the negative ones \ and theti add the two terms according to the rule. Thus, 11. Add 4ii*-|-7fl* — 3a* + iau»* — 8a*-|-«' — 5^* to- gether. first, 4d*-|-ya*4-i2«*-|-fl*5=:24a% the sum of the af- firmative quantities, And — 3a*-*-8fl**— 5fl*=— -xfia*, the sum of the nega- tive. Then 24^i*— i6fl*=?8fl*, the. sum of l|ie whole. 12. Add 5fl**---4fl**-|-ioa;tf*----8jx*--H$dx* together. First, 5fl;c*-|-iOflflff*cii5fl*f*, And — 4ax*— *8fl;^*— -6flX* = — i &ix* ; Therefore the sum of these quantities is -|-X5ax^---i&i ^»3- 14. — 6y/ax — 2J? ^\^^aX'^ + Z^ax + > + «** — 6^ax —77 — 3«** -|-io^«« +57 +3"** L L (;as& 274 CA8B III. When the fuanHlUf an unKle. RULE* Set them down in a liae« with their signs mi coet*^ neicnts prefixed* EXAMPLE. To 341 —Sat; +2v/«*— ** Add a* +5 j^ — 9 Sum 3^+2^— 8«y-|-5j^.2y^fl*— ^*— 9 CTMSJt £ZAMFL£S IN AM>ITJON. 3** t— 9*' +9^^ +^;' 7fl* + 5«» +&* —6 Sum 155* — li^+3x/ai+6c* — sd+xy — 2j;^+4* 2. * Here the first column is composed of like quantities, which iu« added together by case !• The terms — gP and +9^' de- stroy one another ; and the sum of — 12^* and +5^' is — 7^% by case 2. The sum of +'j^ah and — ^^aB is +$\/ai. In like manner, + 10 and — 6 together make +4 ; and the rest of the terms being unlike, they are set down with their respect- ive vgns and coefficients prefixed, conformably to case 3. SOVnUkCTIOK. 2, 3- 4- ♦ — 2x*y «»^ 8+**— a 10— *«—«•— ^ 2^5 SUBTRACTION, RULE** Change each -j- into ^f— , and each— into +/ '^^ the subtrahend^ or suppose thtm to be thus changed ; then proceed as in addition^ and the sum wiU be the true re* mainder. - EXAMPLES. * This rule is founded ou the consideratioo» that addition and subtraction are opposite to each other in their nature and operation^ as are the signs + and — y by which they are expressed and rep- resented. And since to unite a negative with a f^oiititot quantiuf of the same kind has tlie effect of diminishing it> or subductin|[ an equal positiye quantity from it ^ therefore to subtract a positive, which is the opposite of uniting or adding^ is to add the equal negative quantity. In like manner, to subtract a, negative quan- tity is the same m efiect, as to add Qr~ unite an equal positive quantity. So that, by changing the sign of a quantity from -J- to — 9 or from — to +, its. nature is changed from a subductive to an additive quantity ; and any quantity is in effect Stt|>tracte4 •^by barely changing its sign. * . « ..* EXAMPLES I. 2.-3. 4. 5,- From 5» 3*. +4* — » r +8** Take 2« 5^ —2* +fc — 8^' Rem. 3a "^TM +6* -—9^ +i6fl* <5. 7. 8. 9. From — iWr +9c* — 4^* — 7** Tike . + 8*r +8if» +2rf» — s*» Rem- wi6*r ♦ — (Srf* — 2;if* ' ' ■ II MP I' T f i »*^ ZO. XI. 12. From — I7xy 5^^ 5^* +3*5?+ 14 Take — ^12Ary ltfAr'4.4 fl*+2*;C lO Rem- * 3«^*— 4* 4^*+ *x+24 ?3t * The ten foregoing examples of simple quantities being obvi- ous, we pass by them ; but shall illustrate the eleventh example, in order to the ready undersunding of those, which follow. Iq the eleventh example, the compound quantity 2^ix*-f'4 ^^"g ^k- en from the simple ouantity 5£mp*, the remainder is 3dwc* — 4, and it is plain, that the more there is taken from any number or quan« tity, the less will be left ; and the less there is taken, the more vrill be left. Now, if only 2ax* were taken from 5^5? *, the re- mainder would be 3^Me* j and consequently, if 2ax*+4, which is greater than zax* by 4, be taken from 5/ix*, the remainder will be less tlun j^jjc* by 4, that is, there ^yill remain 3^1^?* — 4, as above. For by changing the sign of the quantity 2^a:*-}-4> and adding it to 5/7.X*, the sum is ^ax^ — lax* — 4 ; but here the term — 26X* destroys so much of $ax* as is equal to itself, and so r/7x* — 2ax* — 4 becomes equal to ^ax^ — 4, by the general fulf for subtraction. MULTXfLKUTlOlS *?7 13- '4- -•:':iS- ts. from 9y^flflf— 5/1 6^a*^y ' '^ v*-f*- Take fiv^flX 9\/^*+** — y« x/x+y* P.em. 3v^a« — 5« — 3v^fl'4-** +5* %/*+*•-/ 16. 17. ^18. — 3«'>+i V*)+2+*';'- 6*'— io+4i— «» 19. . 20. «i. 3x^—20 4;c' — 3X^4-* jcjr' -|-i<w^«rj,-|-io 4^;^ — 3^ 3** — 8X^+* ^'j^'-f-^ov^^y+'o MULTIPLICATION. In multiplication of algebraic quantities there is one general rule for the signs ; namely, when the signs of the factors are both affirmative or both negative, the product 18 affirmative ; but if one of the factors be affirmative and the other negative, then the product h negative.* ^:asb ♦ Tbat like sigos make -J-, and unlike signf — , in the prod- uct, mly be shewn thas : " I. men W% CASE I. ^Tien both the Jactars are simple juantitief. EULB. Multiply the coefficients of the two- terms together^ to the product annex all the letters of the fetms> and prefix the proper sign. EXAMPLES. I. 2 3- 4* S- Multiply a -3^ j^ —Scd •—a by h — 2u: 3 . ~4« b Product ab +(Sfc liah ■\-iocdx -—ai . 6. Multiply !• U^hen -^a is to be muhiplcd fy -f"^ ^ ^^ implies, that -f*^ i* to be taken as many times, as there are unics in ^ ; and since the sum of any number of affirmative terms is affirmative, it follows, that 4'^ >^ +^ makes -{-ab. 2. Wben two qwmtkies are to be multiplied together ; the result will be exactly the same, in whatever order they are placed ; for M tiroes b is the same as b times a ; and therefore, when — a is to be multiplied by +b, or +^ by — a, it i$ the same thing as tak- ing — a as many times as there are units in +^ ; and since the sum of any number of negative terms is negative, it follows, that -^flX +^, or +^ix — b, makes or produces — ab. 5. If'ien — a it to be multiplied by — b ; here — a is to be sub- tract^Ras often as there are units in b ; but subtracting negatiySs is the same as adding affirmatives, by the demonstration of the rule for subtraction; consequently the quotient is b times a^ or +ab. Otherwise. HVLTsnio/i^iom. iProduct •— tSfc *— 13U»* "^z^dx ^ , lb. II, Multiply *^— tf* 4"S*y '^^^xyz by — 7^ *— 3 ^^^^x Product NoT£ t. S0 multiply any power iy another of the same root ; add the exponent of the multiplier to that of the multipli- candy and the sum will be the exponent of their product Thus the product of a^ multiplied into a' vit^a^^\ or aK That of yinto iv is a?"**. That of *" into ** is ;v"**. That of ic^ into *" is •*♦"• And that of cf^J^ into y"^ is cf^"^*^, or ^«*^. Again^ the product of a-f"^! multiplied into a-f-x is «+*! • . And that of x-^-yi into x-^^yX is ^+j^| • This " 4. ■ ■ Otherwise. Since a— a=Of therefore ii — aX— ^ is all^so, beause o maltipSed by aay quantity is still .0 ; and.siaa| the first term of the prodnct, or « x^» ^^ by the second case -; there* fore the last term of the product, or — a.X"-^ must be +#1^ t6 make the sum .;=:o, or -— «^-|-a^=:b ;'that is, — « >4-^s=+«^* *♦ tto ALGBSBA. This nde is WMl^ applicable, whea^He^exponents o^ any roots of the|iK quantity are fractional. ThuSf Am product of a* multiplied into a* is a*Xa : In like manner, x^X^e^Xx* = x^'^'^^^ = x^ = ?c' Hence it appears, that, if a surd square root be multi- plied into itself, the product will be rational ; and if i surd cube root be multiplied into itself, and that product into the same root, the product is rational. And, in gen- eral, when the sum of the numerators of the exponents is divisible by the common denominator, without a re- mainder, the product will be rationaL « Thus, a*Xfl*=:fl*+*:iia""^=/:tfl'. t Here the quantity a^ is reduced to a^, by actually di- riding 8, the numerator of the exponent, by its denomi- nator 4 ; and the sum of the exponents, considered mere- ly as vulgar fractions, is -^^+^^^^22. When the sum of the numerators and the denominator of the exponents admit of a common divisor greater than miity, then the exponent of the* product may always be reduced, like a vulgar fraction, to lower terms, retaining still the same value. Thus, x^Xx^=zx9zzx^. Compound surds of the same quantity are multiplied in the mfixt manner as simple ones. Hius, ii+*l* X tf + «l- = fl+^r = ^+A = ^+* » • Bo likewise y/a^^x Xv/^+* =;||jK&«|"^ =:a-|*^ ^' And y^^4"^ X V ^•h* =V^fl4"^ =:flf-f-* *' And y/a^x X\/^+^ :=za*\-x. These examples shew the grounds^ on which the prod- ucts of surds become rational* Note 2. DifFcrent quantities under the same radical sign are multiplied together like rational quantities, only the product, if it do not become rational, must stand un- der the same radical sign. Thus, \/7X\/3~V^7X3=\/2i; ^/^cx X\/2y :=\/i^cxy . And ^]|7xv^^^==v/8r^/'=:8ci*!". It may not be improper to observe, that unequal surds have sometimes a rational product As v^SiXv/^ =v^64==:8i \/!>^^y y.y/xf -zz^.x^ y^zzzxy. And ^a+xf^ X v^fl + icf X y/a+xf^ '' = \/a+x " =a-|-^. QJLtt M M 8ft > MLOLBfJL. "• %f\ OASi II. Wien onHf^hi j^ctors is a eompound quantity. ' riTle. Multiply efdj term o^ the multiplicand by the muht^ plier* ftXAMPLBS. 2. Multiply 5tf+fc by y 2\/ ab -*-4* • '^'jc^ax 7i Produa X5flr-|-3fc* 6b^ah — %h^ -{-l/^c^ax mammmmmm^m Multiply 3flf— 4^ii+5^y/';c*— ^* -— 6i^<: by 2a^c Product 6ax*^c — -g^y/r/i + 1 oad^cx* — 9*— 1 %abe^ Multiply -if — 8+2^7 by yjr 5- 2**+*' Product Multiply by 6. — 2*» Product 2j?*— 8;c*— ^x CASB MULTi?UCATiON. - 2$3 CASE III." j^:i^ JVken both the factors are cofgpouiM'quaftfitus, fLULE. Multiply each term of the multipllcahd by each term of the multiplier ; then add all the products together^ and (ihe sum will be the product required. EXAMPLES* Multiply by Product fl*+2^^-p*^ a* ♦ —A' 3* Multiply * In the first example^ we multiply a+it the multiplicand^ into ^f the first term of the mukiplier» and the product is a^+aB ; then we multiply the multiplicand into bt the second term of the multiplier, and the product is 0^4-^*. The sum of tl\^ vuck products is a*+2ab+B*y as above, and is the square o£ a-i-L In the first example, the like terms of the product, viz. ab and «^, together make zoB ; but in the second example, the terms +ab and — ai^ having contrary signs^ destroy each other, and the product is «* — ^*, the difference of the squares of a and b. Hence it appears, that the sum and difference of two quantities, multiplied together, produce the difference of their squares. And by the next following example you may ob. $erve, that the 'square of the difference of two quantities, as a. and kf is equal to a* — 20^ 4*^ ^> ^^ ^^^ ^f their squares minu!^ wee their product. 184 A^^UMJU. Multiplj by Product 3-T I ac ' ie ■ iJI Multiply by a^ t/ ah a-j^ j/ab •\-^ab^ -^s^ab — A Product 7- fl * — ^ Multiply #* by . 7.V— 4 2;— 3 I4>9' — *Zy ' JPfOduct ma;;'— 2i;v— 8;-|- 12 l(. Maltipl|r mrniojx. 085 8. Multiply flf'+ioxy-f-? , by ;c*— 6;vy-f-4 — 6x^y'^6ox*y^''^42Xy . +4^* +400:;;+ 28 Product ;v * -|-4Ar ^y — 60X ')> * +^ ^^ * — 2;c)i+28. 9. Multiply A;^-j-;v*ji+xy'-f;y' by ^—y. Ans, Af*-|— ;i*^ 10. Multiply AT* +X)'+;^* by x* — xy+y*' Ans. x^'{'X*y*'{-y*. 11. Multiply 3;^* — 2xy+5 by fltf*+2A?y — 3. Ans. 2^*'{'^^y^A^^ — /\x*y*'j'i6xy—JS 12. Multiply 2a* — 3^i*-f-4Af* by ^a^-^-SaX — 2a?*. Ans. lo^j*— 27«^flf+34/»*;tf* — i8tf^' — 8;^^. DIVISION. % Division in Algebra, as well as in Arithmetic, is the converse of multiplication, and is performed by beginning ^ at the left hand, and dividing all tlie parts of the dividend by the divisor, when it can be done ; or by setting them down like a vulgar fraction, the dividend over the divisor, . and then rjpducing the fraction to its lowe$t terms. ? In division the rule for the signs is the same as in multj^ jp plication, vix. if the signs of the divisor and dividend be ' . alike,^ ?^ • - C^ l86 AI.6URA. a^ke, that isi both -j- or both — -^ then the sign of the quotient must be -|* > but if they be unlikei the siga of the quotient must be — .* CASE I. When fie divuw and dividend are both simple quantitiet, RULE* 1. Place the dividend above a line, and the divisor un- der it, in the form of a vulgar fraction. 2. Expunge those letters, that are common to the divi* dend and divisor, and divide the coeiEcients of all the terms by any number, that will divide them without- a re- mainder, and the result will be the quotient required. 3- ahc bed ^ . l8x' I2ab abc a Quotient — =z=2^ — = —4/16 7~,= •; ^•9;!; 3 -bed d EXAMPLES. Divide by z. 9* 2. — ^2^l^ 3 Jk 4* Dividt * Because die divisor, multiplied by the quotient, most produce the dividend. Therefore, 1. When loth the terms are -{^ \ ^t quotient must be 4-, be«« cause -f in the divisor X + in the quotient produces -f in the dividend. 2. When the terms are both — ; the quotient is also -f-, be- cause — in the divisor x 4* ^ ^^^ quotient produces — in the dividend. 3. Wbesi 1 A SITISIOM. 4- s- 6. Divide ai —15 'jabcx^ by 2b 3" Sax^ »»7 Quotient— =:-s:4^.*^-iti'*- ^ «i-~ 2:^^ ^; Divide i6flf' by 8^. AnSi 4V* 8. Divide 1 2^*^' by 30*^. Ans. 4X. 9. Divide — ^5^;* by 34^. Ans* 5/. io. Divide r^i8aAr*jp by ^'^Saxt. 42 It may not t)e amiss to ot>serVe, that when any quanti- ty is divided by Itself^ the quotient wiH be unity, or i ; because any thing contains itself once : thus X'^X gives tp. and -y/atf^ divided by v^aaij gives i. Mote i. To divide any power hy another of the same root s subtract the exponent of the divisor from that of the dividend, and the remainder will be the exponentlof the quotient. Thus, 3. When me ietm is -f and the other — ; the quotient must be >-<-, becaase 4* in the divisor X — in the quotient produces -- , in the dividend j or — in the divisor X + in the qaodent gives — in the <fi?tdend. r So that tibe role is general ; like agns give -|-t ^md ualike \' , i^gns give — , in the quotient. a88 ALGEBRA. ThttSi the quotient of «• divided by «' is a*""^, or a^. That of x'^hj xisx'^^ That of 9^ by je* is Jtf^*. That of ***" by x" is ;tf*. And that of x" by *'' is «f"^. But it is to be observedi that when the exponent of the divisor is greater than that of the dividend, the quotient will have a negative exponent. Thus^ the quotient of x^ divided by x'' is X*""^, or fl^*. And that of ax* by ;i;^ is ax''\ And these quotients, viz. ^"* and ax"^^ are respective- I d ly equal to — and — j for x^ being actually divided by X X X^ I * . . t . ^** ^ X^ gives -y sr-y i and tf;c divided by x^ gives — j=z-j^ as above. ax^ a In like manner, /wf" divided by ex ** gives — r- ~ — - . And the quotient of fl*+itf* I dividSd by fl[*-J"^*l -J^* «»+JfM . Moreover, a\ divided by «* gives a* ZZo^zz^a y-* «+xl'^ divided by fl-|-;i;l^ gives fl-4~^l^ = And «H**r divided by «* + x*r gives«*+x*|" . SCHOLIUM. t When frdciionii kxpomnts rf the povfirs of the saM fM fki^ MtTA'ifmu AnemttuttoTf i^^ to a common denominator^ like vulgar fractions, and then their humerator^ may b^ 4ddcd> or.subtracted, as beifdre. V v. . - - '•• »_ . . k V Thus, the t^tiotlent of or+^l* dividci by ac-^-xX^ is N6TS 2. 3urd quantities under the same radical ^ign are divided, .onf by &e other, likr rational quahfitles, only mt quotient^ if 'it 3o not become ratibxial^ must stadd un- der the same huliod sign. ThUSi ilijbe qi^tient of ^%i divided by y^j h y/y* That gT ^tff by ^^ is y^^"* " That of y^i6c by V'^ac is.^S, or 2; r Thit of \/;^{' by ^i^i" is |. ^ And that of lOflV/l* by 3aV>^{* is 4% •J* J ^ feAS« lii-./ ^ ^ ' ' W^iM^ f^ diidsor i^ a simple qudntitj mtd He dinfidefd a tom^ ^<mnd puMsty. . - ^ ' ktTLfi. ' 'Dividife 0fevficxxd ^ the dividend by ^ divisor, as in r the first case* , . . . ^ '! • EXAMPLES* I. ' 3^)*5^+3MsH^ quodcnt. tient 3. Divide 3x*«^i5-4-6«;-{-i3a by 3;?. p a 4. Divide 3fljg4-ia<rf< g ^' pg *^ by 34it« Ans. ^-f-4;c«*— 3<y* 5. Divide loa*^— 15«*— 5^ by 5^?. Ads, Otf*-*— 3«— I* CASS III* ■ » ■ ... JFien thi £visor and divnknd are kpth confound fuaniitkr. 1. Range the terms according to the powers of some letter in hoA of them^ placing the Ughest power of it firstj and the rest in order* 2. Divide the first term of the dividend by the first term of the divisorj and place the result in the quotient. 3. Multiply the whole divisor by the quotient term> and subtract the product from the dividend. 4. To the remainder bring down as many terms of the ' dividend as are requisite for the next operation j call the sum a diyidual^ and divide as before ; and so on^ as in Aiithmetig. SXAMPLSS* ■■■•?;'■ ■,•.•..- . ■ ■.. Sr«-AXVii.a!^ .■ ♦ i^i . Vf ' ; ■••t . i. lit it ie ^mre^ » jfifia^i V \fj ax\--x. " _"_^ — 4^ ^itr^fap^ ^ ftSt dividual •^— — r"*-^ — ' ^ J^ ax^-j^x^ second dividtal. , m ♦ ♦ \* - ', » - I • ■ ■ ■ ■■f a. I)ivide ^ The process may be e3cpUuned thiis': First, tf' divided by if gives a*. for the first term of tbe quo^ ^C^t, by which we maldpty the Whole divisor, yiz< a+Xf and ihe product is a^ +Vi**, which, being taken from the two first terms of the divideijd> leaves — 4/1*.^;- to rfiis remainder wc iffi&^dowh ^^^ax^ the next term of the dmdead,..aad the sum Iji — 4a^»! — $ax*f th« first dividual ; ngw. dividing — 4a**, the first term of this dividual, l>y 0, the first term of the divisor, ther^ comes out -^4/i«i a native quantity,, w^tda^ we, also put in the quotient ; and the whole divisor being multiplied by it, the prod, net if *-4i^x«-44X^, which being ul^en from thd first dividual^ the remainder is +<?x* j to which vc bring down «', the last i term of the dividend, and the Sum is ^-^w'-|-^S ^'e sedond dt- fidual ; and +ax^» the first term pf the second dividual, dividr ed by ^/, the first tcrni of the divisor, gives pe* for the last term p( the quotient ; by ^hich we multiply |h^ whole divisor, and the product is -J-^ix*+*'> which being taken from the second dx- ^^dual' leaved nothing ; and the quotient requn-^ is a* — /^x-^'X*^ ' r:S:.- ^9^1 9« 4^)8aiv/ tient. 3* Diride 4. Divide * Here tf» the fust term of the dividen^i^ tciua divided by ^at the first term of the diyisor, gives \fa -for the 6rst term of the quotient. Fora=rii', and i/a zs a^f and the difference of the t:sponents is i — f, or i ; therefore a^ divided by /z* gives 4* * = ^j* = v'^* *5 ?.bove. Or it niay be considered thus : ask what quantity being multiplied by ^a will give a, and the answer is v^a 5 then the divisor being multiplied by ^a , the product is a — j^ab } but diere being no term in the dividend. --^ — . '• * ■ J ' " ' ■ ' ' ' Remains + ac* — c^ Here it is obvious, that the divisioil caiinot . terminale without a remainder 5 therefore we write the ijivisor tiridcr the remainder with a line between them, and add the fraction to Of^^^rCj the other twa terms, to complete the quotient. . . > But when the dividend docs not precisely contain the divisor,, then we generally express the whole quotient as a fracticjh, having reduced it to its lowest terms, or rejected the letters and factors, that are found in every term of the ^vidend and divisor. 5- Thu^ that corresponds to "-^t^ab , the second term of this product, we subtract a^»/ab from a^^h^ the dividend, and the sign ol the quantity — ijah being changed, the remainder is + ti/ah — h* ^ow + »/ab , the first term of this remainder, divided by i>/a , ^ the first term of the divisor, gives — »/h for the second term of the quotient, by which we multiply the divisof, and the prod- uct, viz. J^i/ab — h, being subtracted from the aforesaid rof. fpiainder^ ^lothmg remains ; and the quotient is \/tf-i- v^i« 5. ThatrS^U-^H^-^-axf divided by adx-i^aaH • -^(bc+anx </+» Here the quotient ■ ] , i$ reduced to ^ odx^anK ah^cX+X* , _. ... \ • ^ V ■ ■ J by dividing every term of its numerator and denominator by ate, . . 6. And fl4-fli-W* divided by a*— •arr-J-^*^* gives Heve the quotient cannot be reduced to lower terms^ be< cause the factor a is not to be found in the term rf*. But it is to be observedi that though a fracti(m cannot be Induced to lower terms by a simple divisor^ yet it may sometimes be so reduced by a compound one ; as ixgU ap^ pear in the reduction of fractions. . 7. Divide a^'-^x^ by fl+^. Ans. a*~tf;c-f-;tf\ 8» Divide a^— 3^*>-f"3^J'*""J'' ^7 ^J"— y- Ans. a * '-^Zay^y*. —9. Divide 6x^ — $6 by ^x — 6. Ans. 2x^-^4x*'^SX''^i6, 10. Divide a*-— 5a*A;+iOrt^Ar' — lOfl*^'+5^i** — A?* by ii*— 2^x-f-»v*. Ans. fl^— 3fl*«+3fl'4;* — x^{ FRACTIONS. Algebraic Fractions have the same names and rules 6! o|>eraiSon> as fractions in Aridimetic. fEOBLEM PROBLEM U ^ To jind the greatest common measure cf tht, terms ofapJiUim% ttULE* 1. Range the qnantitjes according to the ^i^n^i\a o£ Some letter, as is sKewn in division. 2. Divide the greater term by the less, and the last di- vision by the last relnainder, and so on till nothing rc^ main \ then the divisor la$t used will be the common measure required. * ' ' Note. All the letters o^ figure^, which ate, common to each term of any divisor, must be rejected beJTore such di- visor i$ used \tk the operation. ' EXAMFLC^. ^ 1. To find the greatest common tti^rtiUrd rf ■ '/] > ■, or <-|-;e)fia*-f a*«(a-*' ' ■ Therefore tlie greatest common meastire is c-{^. 2. To find the greatest common measure of ■ . , .*... . or •+*• )»'+2*«H-4»(«+* Therefore fc+i is the greatest commoi a* To fiai l|» grditet liotiuaoll inettttre ■ ? • ^ 4* ^^ fl^ ^'^ jStsActk cottuiiW iiMMufe of ■ \ ,/ ■ « • Axis. Jr'-|-*r if^ ndiwir abaction io Us lovfeit terms. > 1. I^ the greatest . common measure as in tfac Jast 2. Divide both the terms of thxi ftftcdon by the coni^ men measure -tlitts found, and it wid be reduced to its lowest terms. BXAMPLtS« X. Reduee — ■ , « to its lowest teAns. or c^ >fl*-j-fl*x(af >■• • ii I ' Therefore r-f-^r is the greatest common measure $ 9Qd tf-J-^) -T^T^C "T *^ ^^ fraction required. t. Hating ^»,^^.i» - P^cp, it U rc^pw^.«o.f«4ac4 . St to its least terms. Therefore ir-|«5 is the greatest common measurei and ir+J) f . , 1 / ■ ( *'7v U the fraction required. 3. Reduce -^ — rr-r to its lowest terms, Ans. -—t— . I . » . - . 4. Reduce . -^^ to its lowest terms. Ans. ■■ X*— >* «'+j^* ^4 3p4 S- Reduce , . . , , to its lowest terms. t ■ ■. .... Ans. ■ ' ■', ■ ".. ■ 5*-* PROBLEM III. 7(9 rA/»^ a mixed quantity to <m mpraper fractm. RTJLE. ^ Multiply the integer by the dcnombator 9iM^ fraction, md to the product add thi^ niUQcrator 1 then thlif ^^namL* ij^iiator O o nator be>iig,I^ieed under thi^ 8am^«|U.Qfe die iinpropet' BXAltPLBS. 1. Reduce 3-f to an improper fntq^a* . ' • .,1 . 1 3l.=m±£h^;i^k±Lsd the ans^n •'7 7 7 7. • t « 2. R€ii|Me, A-t'-^^ttin^Fi^pcr fincdon. , . a - " » i — — = the answcn 3. Reduce ^r^";^ to an impropex: fraction.^ 4; Redute 8^ to an Improper fraclita* ' •. .An& -*• ' ' ' ♦ . \ - 5. Reduce i— — to an improper f'raction. * a— iy Ans. ■ 6. Reduce x to arf improper fraction. 7. Reduce lo-j to an improper fraction. PROBLEM IV. To reduce an improper fraction to a nvhole or mixed quantity* RULE. Dinde the numerator by the denominator for the inte- gral part I and place the remainder, if any, over the de- * nominator '^ « fipminator for the fractional part ; the two joined together 1^ be die mixed quantit]^fec]iUred. .. |. To reduce y to a mi«e^' quantity. y = 17-7-5 =3t the answer required. [\^i fied^ce .m ■ tp a ^ple pr |?iiji^4 qus»ti^ — - — :=zax+a* -r-Xi=tf+— answer. 3. Reduce — — r to a whole or mixed quantity, - a^ — a* ■' ■ - . ^ ,. . ja* r— X — >;r/«A-r-tf* -4-^ i . n . answer. 4. Reduce ;.if"|" -^ ^ ^p a whole or mhcedjqusintity, -i^ =^ =±: ay-\^2y --r^4^3:;>+-=w answer, t. Let ' • - be reduced to a whole or mixed quan- ' ■"■ ■' ■ . - . . ^ .. • ^* ^ty. 'Ans. 3^ — — , a ^ Let — ^ — be reduced to a whole or mixed quantity. Ans. /sf+.^H — ^ 2X' ..3_yB 7. Let ' r be reduced to a whole or mixed quantity. PROBLEM M' Multiplj <adi fliinMMftqr into all dit (icm^mUuitoirs aevw tJic dfiBomuuton together for the commoii denonunatos^ y. ReiS^ee — and — fo a cbSnmosi deuomiaa^^' ^J^? theocj^ Jyjirn J tlie a«* muwatbrs. ^X^ r:? A^ the eoniitab&'deiiapai^iatm Therefore r^ and — appr- and j-, rc8pecfwipl(r> the frw« tions required* t^ Reduce -7-, — and -r-^ to a common detiominatort PC a by(Jby.d z=z b* d > the numeiatow. hyicy^d = bed the common denominator. Therefore -j,^ and ^= _, ^ syul ^ respective, ly, the fyactipns required, J. Reduce |. IlecliKre -^ «nd «-« to eqmraknt f|!SictieQ% hmn^ m common denomiaator^ Ans* — — and — ^ ac Of 4* Reduce -— and -^ to fracdon^^^ having a coi^imoa denominator* AdB. 7- and — I — , (fc PC 5. Reduce ^, -^ and d^ to fractio^Sj Jbiaving a common denomiiiator, ^ ^^^ . . ^^ . An3- 1^, 4^ and ^ 6. Reduce -^^ — and ff+*^» tp fractions, having a 45 coq»moa denoxninator. ^ ga tax . i2ar+2Ax ^ jMs. -21. and ^ ^ t PROBLEM VI^ To add fractional quantities togiftber^ EULB. 1. tleduee the fractions to a conmioii denominator.* %. Add * In the addidon of mixed quantities, it is best to bring the firac^onal parts only to a common denomioator, and to afBst their Stttn tatiie sum of the integersi interposing the proper si^. 2, Add all tbe numerators together, and imder the suii^ write tlie comiiwn denominaftiry and it will give die stm^ of the fractions required. ^ZAliPLBS. If. Having — and ^ given, to find their sum. Here *^X3— 3*? the numerators. And 2X3=^$ the Common denominator. Therefore ^r+'r^^^ ^^ the sum required. a Q q ' ' ^. Having -J-, — and -j given, to find their Sttm^ aXdXf z=z adf^ ^ Here c)<bX/ = ^^ V the wm^ator^. ^nd by.dXf = ^^ the common denomln'atoi. Therefore .^+^+~ = ^3:3^±_, the sur^ required. 3. Let a . and i-j- ?^ be added together, ^ ^, ^ . > the numerators^ And bX<^ = ^^ the common denominator. Therefore «~ 1^14-^4.-^ =«-3|:+^+igl=«+ b4 ~— the sum required. SIUICTIONS; 303 4. AUZ and 4. together; ' ' Aal IZi^. 2d 5 loi c. Add --, .^ and — together. Aito; ir+4?- or i^;^. * 3 4 ' 12 12 ., ;($. Add ^^ and ^ together. : -"- Ans. ^trj±. 3 7 ^ 21 7. Add ^+^^ — ^ to Z^'^'^ — ^- Ans; 4;^-} 3 ^ ■ 4 " PILOBLfirt vit. To subtract one fractional quantity from another^ RULE. u Reduce the fractions to a common denonunatory as 1^1 addition.* ; i. subtract one i^umerator from the other, and uilder their difference write the cominon denominator^ and it will give the difference of the fractions required. EXAMPLES. . !• To find the dificrence of — and — . . Here ^Xn rr iix? .t O . > the numerators. 29cX 3 = ^J ' And 3 X 1 1 =f 33 the comnit](n denomiinator. Therefore * The same rulejnay be observed for mixed quantities in sub- traction as in addition. 4k I II ' ■■ I I till III r I i| muU And 3^X5^ ~' XS*^ the common denominator* the difierence leqiured.. ' 3* Required the difierence of ^ and ^. Ans. «^^« 4. Reauired tbe difi«retM)e of ^ and ^ Ant. -^* 5. Sttbtiact 4 from 2±?. Am. '''^'t^ -' 6.Takei^from 5i±l. 4X3* 7. Take ir--*~i from 3*+j. Am. «H ^^^* PROBtBM ^^ VRO^LB It vin% Multiply the humetatots together ftit a new v^ummm^ ^itd the deadminatots for a new deaoodostor ; and it vUl gift iii$ prodi9u:t required^ "" .{ ' . ■ . - - . Jft^ Required to fiiid the product of •-?- and — ; * te ^. Required the product of — , 3- and — — -. • ■ * Here ^il4f211£f ^1»f!«ifl ^ ^„^ ^^j„^ 3. Reqtiked the ptbd«ct <rf -2. ad* 5^* Here ii>?4^=fL±2f the produi* feqijiired- 4. Required ., '■'*■■■■ ■ I -- '^ . ■ . ■. .. - "* i. When the nuiuerator of one jfraction^ and die debominat* tft of the othier» cati tife dfi^ed hy s6nji& ^(n^^jr* Which ^iBCodOh «bn to both, the quotierits may be tfsetf bHftttiid' of thw. ' " 3. When a fractiotfls-to tfe nUlhipUeif (i/'jij 1^ t&C JtfW- • Mtt is fmidd by multiplying iht hutrictitdir % ftj jiiff if the^imc- ger be the ^ame with tlie deiiomiDkory the^omtritto Oliy'be ^o en for the product* * ' • '' • 3. When a fractroft is to be ttruhlpH^ Iqr jrtiy^qaaBUty, it h die same t^ing, whetiier the triUffiitator be ti^pKed by It^ m ^c iienbmkiator divided hy lit .' * ' 4* Required tfic prisduct of — ..W y. Ani. 2!^; 5. Reared tbe product 6( ^4 — and — . \^ ^ Ans. — ^— • . ., It t. Required the product of % and * "^ > Am. y-^ 7. Required tifie product of x^ -i- and -"ti* Ans. PROBLEM IX* 51? divide one fractional quantity By another* Multiply the denominator of the divisor fey the niftnet-* ator of the dividend for a new numerator^ and the nit- merator * I. If the fractions to be divided have & commoa denoau* aator, take the numerator of the dividend for a new numerator, and the numerator of the divisor for the denominator. 2. When a fraction is to be divided by any quantity, it is th^ same thing, whether the numerator be divided by it, or the d«- ttominator multiplied by it. » 3. When the two numerators, or the two denominators, can be divided by some common quantity, that quantity may be thrown out of each, and the quotients used instead of the fractions first propostd*^ yv^erator of the dxviaor by the ^^nofpiasitof of tbc dividend for a new denominator. • - ;•* - • - OXf invert the terma of the diyiifjri js^ di^ )ji])ldply |>y it^ exactly a» in mu|tipBcatipff;; _^ '^^\ ^^^ . 1. Divide JL U it ? ^ " : / Hcfe Y><£;==^=T= ^T '^ ^« S»?tknt ipccj^irc4.^ *' _ ■ ; N 2. Divide -7^ by ^. Wre ~x— =1— 7— =— y w the quotient required. 3. Divide ' , by — -7—, ip^quired! X. iSrvidc -1^ by ^x. , ^ns. 1^. ^ , 7 35 p. Divide — i— by — • Ans. — ^ . ' . ?• Divide ■> .-41% f« *: lO* ■•■•■.f,.4j^^. '. »t V. M- r,". • . 1.. , ^Iinpos.imoH is the condnual inn|&>Kq}fioi:^ of a qusnitiv if'mto Itself, ailil die ^iroduc^ ^c^eerai^li^ called iiicfamferi of that quantity^ aod the quantity 'itself i$ call- ed the roqi. Or it is the nftthod ^£ %dteg ^.^quar^ cub^ l»quadiKate» ^c. pf smj giveu'^uaati^ Mi^tiply the quantity into itself, ^ the qiiantity he taken for a factor as many tisias as,tfacse-jSre wits in thQ index, and the bst prodact V^U l?e the |k>iv«s re^^iifedt ^ •■'-0% ' ' ...-- -^ .. Multiply the indei; oS iK^ quantity hfiBbit iTncitor^thQ power^ dxki the result will be the power required^ PXAMPLI^ ■■ " ' > ■ m * root a* root s\ 2= square <»* -."PB squaro «' ^ tube a^ ■?; cube m* = 4th power a^ ^ 4th power •' = 5th power. ^^" i=? jthpowpxu ^yh * Any power of the product of Jtwo or more qoanutiet i| f qua! to the same powers of the &ctors^ nmltiplidi together. And any power of a fraction is eqaal to the saSM power of tt^ UViUcr^^or;^ divided by the sam^ power of tbe de '■4 INFVJ.OTIPN, wm* ^^a^ == cube mmm2^la^ 5; 5th pOW^F, ~ 8^*«f* =3 cube — *'3aa*;ip y*j»«> — 5th povwar. W I M l ■ . ^ f l i I ■» '* -1..'^. ' . . g '■ ■■ ' ' ■ * '- '" i ^'^+4^y^-|:6«*x*-f-4rt^;v+fl* =j 4th powen The third power of x^Mj;-^^r or ;^^ The fp!jM:th pQWCT of 9aH* :i% ^*^^' '^^ or ^]6<^! *i*. :1;Tiq iwth:powcr of y* is>T^ Tbe^econd power of <m4* is tf*r j'^^or^j.vl'* that is^, ^^^ ^hc^irtk ptrvrer of-w/jr'' feW" # or.iwc. >Aod Note, AU the odd powers laised finqm « ne^^atfrc^ roo| lore Q^ipitirex and all the even powers arc podtiie. * ' Thi}s» the second power oif --r^ is r!r^X"''«i=^'f hj the rule for the signs in multiplication. The third power of rr« is rf-fl* X '-'""" -J'"'?- The fourth power is -tt^i? XrT<>=+«*- The iMtb power of -r^ is +«^*x— ««t-^*» &^ PXAI^XES FOR r&ACTICB« I. Required the cube of •-•g^'jp^ Ans. -r-jx^^J^^ % Revoked the H^yadme rf - ■ v 3. Requked the 5th power of i^'^if^ Ans. «*r— **X-j-ioa*;«*--i^ioa*x'-f5iix^--^l^^^ SiE ISAAC NEWTON'S Rule Jhr rmsing i bimmal or resubuJ quantity tg any fewer ^ ivbdtever^^ 1. To find the terms fvifbout tie co^ffkkt^. The Index of the first, or leading quanlit}^, begins witl^ that o^ the given * This mle, expressed in general terms, is as fbUows : 2 ' 2 $ ^ivbii power, avi ^ecrea^es . continually by i, ik^ term to the lastY and in the following quantity the i of the terms are o, i, 2, 3, 4, &c. 2. 31? fnd the unick dr coefficients, llic first i^ always \^ and the second is the index of the power ; and in general, if the coefficient of any term be multiplied by the index of the leading Quantity, ^nd the product be divideci by the number oi terms to that place> it will give the coeflicienf of the tctm next following. Note. The whole ntlmber of terms will be one more than the index of the given power ; and, when both terms of the root are +, all thfe terms of the powet will be + j but if the second term be -i— , tiiieii all the odd terms will be +, arid the even termfi — • I. Let fl-j-^ be involved to the fifth j^wer- The terms without the coefficients will be a^y a^X, ^'a?*, a^X^, ax\ x' ; dnd the coefficients will be . . 5X4 '0X3 'OX2 5X1 , or I, 5, 10, io, 5, I ; And therefore ther 5th power is §^^;a^^'^ioa^x*'^i6a*x^'f'S^^^^* 1. Lcfr *', ace. Note. The sum of the cpejScieots, in every power, is equal to tlie number 2, raised to that power. Thus, i + i=ra, for the first power ; 1+24-1=4=2% for the square; 1+^+3+1=: i2r2 ^4 for the cube, or tli^ird power |^ ao4 so oo* * •■^. «*, jt'^ «*i*» i5»i *''A ««», «* } , atid!0ie (Soe^^ts 4lrai be hs, fijs; "^^ ,^^i —^ -y- i dtHl €» xii *»i 'fir, «, ^ J And 4]ietefete' (he 6th'° p6wer of il(— >^ is 3* KbcI the 4^1 power of Xf- hL Aw. «♦*— .4|t*/i4*^*'»'"^'»*** +**• 4. Find the i^th power orti*if^ Ans. /+7«*fl+MJ«?«?-^3S«*/i*4-35«*'»*+2i**«* CtottrctM-iS tHt mi^it$if^ tt kurbte^on, and teaches td find the xoots^of any'giTen ]^wer8« - • CASE U . ikjinijiifro 4u (f ^s^te qumtities* . RU^E.* Ettract the root of tim coeflGicietit for the tiumerical pirti' and divide dic-iiiriioes of the letters by the index of the jfowoTf and It will give the root required. v\•^\^'\ iV^v II ti*' v '•' # ■' • "- -vr i" i ' ' I > ■ ■ ■ '-'^ ^ Aiiy eveh iroAt'df ^n aiErmative quantity may be either -4* ^^ — ? ftW &c ijiiafd" rboi oT +^* is' tithef +11, or -i-^ ; for +a X +<i= +i*,' aftc*J ^ X — «2^+fl * also. * And ^1 i\ ^ \ EXAHPI#E«. : I. The «qaarctoot of 9»»=i 3*^=3^. ^ ' '; a. The cttbe root olr«»* ecox^cr a^r. 3. TKe square itWt of 3a***r=a*»^v^3^iMr'»v^3; 4. ^c dfcibelroDt of ^^^zs^^x^zzr^s^^'^^^^'^S^^*' J, The biqaadrate root of i6it*'X^:s 2 a^x ^ isi2ax^ tXASK IT. ^ojini Ae s^psare^ root ^ a cmnj^ound fuantitf^- 1. Range the quantities aciJording to the dimensions of . %ome letter^ and setthe'root qf die first term in the quo- tient. 2. Subtract the squave of the roott thus founds fron& the first term, and bring' down the two next terms to the remainder for a dividend. 3. Divide the dividend by double the tootp and s^* the result in the quotient* ^ - 4. Multiply And aofodd root of aliy^ qaintity will have the same sign as the quantity itself : thus» the cube root of +a^ is -f^tf ; and the cube root of -^' is -^ii j for -fv^X 4-i?X -H»=+tf' ; and -r-* X — flX— ^=^'* Any even root of a negative quandty is impossible ^ &r netcho: +tf X +«> °or — <iX — a^ can produce — a*. Any root of a product is equal.to the product of th^Mke -ri^ts cl^all the factors. And any root of a fractfon is equal to the- }3ce root ^f the munerator, divided bj tl^ like root of the de« Bomiaator. •'Jk^bBfiBrA. - Mulriply. the divisor' and qiiotient by the term last lUt ia the qootioity &d subtract the product £com the dividend i and so' in, 'as in AritWlJtfc. . -^ .r .■• .. .--EX'i^itpLtfs.*'. *■ ■•■ ■ ' ■•■"* , 4a *4<a*2^:5e^)2|^«^ »4*^^ ^*^ * a. Extract the sc^^uare root of ^♦— -4;^^ +6^*i-^4;j?i|^i 3. Required the square root of flf*'4-4^i^^-|-^*^*-f : ^4* Rcquiredthe*quaw^rootofif*-^2;«?-+- = — """"^T*^* Ans. if *— Ar+J. 5. Required EVOf^UTlON. ^^^[ \ 5. RMuired the sgua^fe:r!^ptof ^»^*7^^ CASE III. ?i jftnd the roots of powers in ^satf^c^L: 'fvt't. - . -- 1. Find the root of the -ftrst termi airf-piape it in the quotient. " « '■' " 2. Subtract the power, and bring dgwn tlfii' second term for a dividend. 3. Involve the jroot, l^st found, to the next inferic>r. pow- er, and multiply it by the index of the given power fpr »< divisor. 4. Divide tjie dividend by the divisor, and- the quotient will be the next term of the root. 5. Involve the^v^jhole root, and subtract and divide as |;)efore ; and so on, till the whole be finished, EXAMPLES. I, Required the square root of a ^ — 2a^ x-\''},a^ X^'-^T. 09; '+. .4^ a' •2a^X'\-ia'x^ — '2aX^'^X\a^-- — ax-^x 2. 2 2a')- — 2tf^V a' — '2a^X^a'x' 2a*)2a'x^ ^' — '2a^X'{';}a'x*- — 2/va;^+a?* # Extract % Multiply the divisor' airf qixotient by the term last in the quoriait, tad subtract the product ftom ^thc dividend J and so'im, as in" AriAirf^ 1. ExtfactJhc squarcLJWt of Tj«'!^iaa^j|-^^ 4/i*+3ax)i2fl*X+i3a*x' •.\. ." .' ■:. > :/ ., ■•• ■1!- 2. Extract the sq^uare root of ^♦— 4;^^ +6^**-^4;ir4^xi — 4*5+4;e* 3. Required the square root of a^ ^ /^^ X'^^a* x* ^ : ^Rcquiredthe*quare^Tootofif*'-^2^?+i— ---^«(-j^. "'■■'•■■■' Ans. **— ^+i- 5. Required \ EVOLUTION. " V;1il ^^ ^. R^uired the sggiaye5r!9ptof^fl^*|-J7^^^^ ^ r. ^ . \ -» -A^^^'^^^lfer-f^ &c. \ CASE III. 3» jff«// /Atf rpafx of, powers in ^ *ijle. ; ' " "^ • 1. Find the root of the -first tcrmi ani^ape it in the quotient. '' « '"^ "*-' '\ 2. Subtract the power, and bring dgwn tHfPsecond term for a dividend. , .3. Involve the jrpQt, last found, to the next inferior, pow- er, and. multiply it by the index of the given power fpr »- divisor. 4. Divide tjie dividend by the divisoi*) and^ the quotient will be the next term of the root. 5. Involve the^ whole root, and subtract and .divide as |)efore ; and $0 on, till "the whole be finished. EXAMPLES. I. Required the square root of a^ — 2a^X'\''7,a* x*~% "> 2a') — 2a^X a^^2a^X+a'x^ • Ta'Ha'x^ fl4_.2a3^-f-3^»^»— -2/^A?^+A?* » 2. Extract 31^ AtCEBRA. a. Extract the cube root of fl;^-|-&c*— 40*54'9^^^''^'^4^ ^6 3x^)6x^ PC^+dx^+itx^-^Sx^ 3flf*) 12*^ flf*-f'^^^'~4^^' +9^^ — ^4 3. B-dtjuired the square root of a*'^2ai^2ac'\-b^'»{-7U 4. Required the cube root of ff^— 6x^-^1 s^"^ — 2QX^-:i\' i»j** — 6x+i» Ans, ;v*-— 2x+i\ 5. Required the biquadrate root of j6a*— 96j';p-|-2i(!j SURDS, Surds are such quantities as have no exact root, beinjj usually expressed by fractional indices, or by nueans pf the radical sign ^, Thus, 2*, or \/2i which denotes tl^ square root of 2. L 3 And 3^, or \/s*f signifies the cube root of the square of 3 J where the numerator shews the power, to which the quantity is to be raised, and the denominator its root. pr6blem SURDS. 117' PROBLEM I. ^ $fi tedf^c^ a r^tiopal quantity to the form of a ^td, |IULE, Raise the quantity to* a power equivalent to that, denote cd by the index of the surd ^ then over this new quantity place ^e radical ^ign, and it will be of the form requived. EXAMPLES, I. To reduce 3 to the form of the square root. First 3X33=3 '=9 > then ^/p is the answer. ft To reduce 2^* to the form of the cube root. First, 2Ar*X2;v*X2rV*=^^*l ;=8x^; - • 3 1. Then y/SAf^, or 8^^| ^ is the answer* 3. Reduce 5 to ^he form of the cube root. Ans. 125!^ or ^125. 4. I{.educe -iyj? to the form of the square root. Ans. -v/-;^*:v*. c. R^wce ^ tp the fbrm of the 5th ropt. , ' Ans. 32p. PROBLEM II. jn? reduce quantities of different indices to other equivalent ones, that shall have a common index* RULE. \, Divide the indices of the quantities by the given in- dex, and the quotients will be the new indices for those i|uantitie8^ 2U Over the said, quantiti^i with their new indices nlace the jgpyen indexy and. thej wijl make ^e equivalent quantities rei|mt^ ' -. Note. A-eommon index may also 1^ found by reducing fhe indices of the quantities, to. a cpmmon denominator^ and invcdvuig each of them to die poweri (|enoted by iUf pumerator* ' \ I. Reduce 1^5"* and ^ fo eq^ihrs^ent quantities, havinr (he common index i* - \ ^-^i2;i;Xi=r|=:i th^ firpt index. i-T-T=fXf ?^i=5.T ^ second inde^ •pberelere i^^i ai^d 9** ' ate fhe* quantities required, 5;. Reduce a^ and y^ to the same common index ^'^ f rT-y=fXf =f the first indi^x. • •J-5-y=^Xf =^ the second index. Tberefore n^F and *'' aye the quantities requited; • i t . , 3. Reduce 3* and 2^ to the commpn index j. * Ans- 27*^ and 4^. 4. Reduce o* and i"^ to the common index -g-. Ans. ^^ and FF. JL 1. 5. Reduce c" and ^"^ to the same radical sign. mn mn Ans. v^tf'^and -^3". 21? reduce surJs to their most simple terms. RULE.* . " - , ' ■ ■ • .'it. ^ind the greatest power contained in the given .si»d^ and set it^ root before the remaining quantities> iidth the proper radical sign between them; EXA^PLkSi i*.iTo reduce i^48*to its»md^t OTtnple terms; .:. answer. i ; 2. Requited to reduce \/ io8 to its most simple termsi 3 *i ' ^ 3 " '3 3 . 3 V^io8=v/27X4=\/27.X\/4=3Xv;4==3V^4 the answer. ^^ 3. Reduce -y/125 to'its tta)$t^im^ktettn^«. . ^ Ans. Sx^S' '4, Reduce \/-rij to itsiAok sitafJle'terriis. Ans. -^v^tS* 5. Reduce ^^243 to its inost simple terihs. Ans. 3^/9* 6. Reduce v^^ to its most simple terms. Alls. "^ \r T* 7. Reduce ^g^a*x to its most simple terms. ^ , ' ^ftOBXEM * When the given surd comwns no exact power, it is already in its most simple termsi IMUlAi ^ipRo^LSM iy« ^ T*' iJd iurd quantifies togetb&v kui,«i t. Iteduce such qiiantitiesi &i tiave tinilke indices^ td oAer -eqiUiralmt tthes, lumiif i gonimda index. 2. Reduce the fe^otif tA a icomifioii dtoominatbf^ smd die quantities to their mosb simple terms^ -' [ 3* Then, if the surd part be the sam^ in ^11 b^ thein^ annex it to the sum of the rational parts with the sign o^ multiplicationt and it will give the total sum required. But if the surd, part be not the same, in all the quanti* ^es, thry can only be added Jbyijefae signs ^ and — * ^ feXAMPLE8» i. It IS required to add i/^^ and t^xfi together. fost, V^7=v'9><i==3v'i; ^ - And ^48 = v^i6X3i=4V3 % Then, ^•3+4i/3=3+4X v'3=:7-/3C=f sum W quired. / - 2. It is required to add \/5oo and ^io8 together. 3 3 3 First, Vsoorrv' 125X4=5^^4 5 3 3 ■ 3 And v^io8=y'27X4=3^4 » 3 " 3 3 3 Then, 5v^4+3\/4=:5 + 3Xv^4=8V'4= sum re- quired. 3- Required the sum of ^^72 and y^ 128. Ans. I4y^2. 4. Required the sum of ^27 and v^i47» Ans. ioy'3- 5. Required the sum of ^\ and ^^* . Ans. ylA^- 6. Required 3 3 3 . •6. Required the sum of \/4o and \/i35. Ans. S\^i^ 3 3 3 7. Required the sum of ^^ and \/y^» Ans. ^\/2. ]^ROBLEM V. Ti suhtractj or find the difference of, surd quantities. RULE. Prepare the quantities as for addition, and the differ- ence oi the rational parts, annex„ed to the common surd, will give the difference of the surds required. % But if the quantities have no common surd, they can cfily b? subtracted by means of the sign — -. EXAMPLES. I. Required to find the difference of ^^^448 and y/i\%, Fiist, v/448z=^64X7=8v/7 5 ^ And ^ii2i=v^i6X7=4\/7'» TlieA 8^7 — 4^^711:4^7 the difference required. 2. Required to find the difference of 192^ and ::^4^. First, 192^ =64X3!' = 4X3^ 5 And 24^= 8X3P = 2X3^ 5 • • Then, 4X3^ — 2X3^=2X3^ t^ difference re- ^ quired. , 3. Required the difference of 2-^/50 and ^18. Ans. 7v/2. £^ . / 2. 4. Required the difference of 3 20 ^ and 40 ^ . 'a 4- - . »ns. 2XS^ R R S- Required AtCBBSA. ^S^ Reqwcd tfae difference of y^j- and v^-j^ Ans. -^ft-v/iS- C Required the difference of ^^ and y^yj* Ans. JLv/i8. 7. Find the difierence of \/8oa^x and \/2oa*x'l Ans. 4tf*— -ZtfxXv/sif. PROBLEM VI. 7(7 multiply surd quantities together* RULE. r. Reduce the surds to the same index. 2. Multiply the rational quantities together^ and the I urds together. 3. Then the latter product, annexed to the former, will give the whole product required \ which must be reduced to its most simple terms. EXAMPLES. I. Required to find the product of 3\/8 and 2^(fc Here, 3X2X^8Xv'6=6y'8X6=6^48=:6v'i6X3= 6X4X^/3=^^3* ^^^ product required. 3 t 2. Required to find the product of xV'y and \^-^* . Here, ±XX^|X W=fX^TT =1X^^4=1 X|X 3 3 3 vi5r=:-j^V' 15=^1/15, the product required. 3. Required the product of 5^8 and 31/5. Ans. 30\/io. 3 3 4* Required the product of ~V6 and ^V 18. ^ ' Ans. V^' 5. Required 5, Rcqiurcd tKe product of \y/\ and -Iv^-rg.. 3 3 .(5. Required the product of ^r« wd 5y^4- Ans. 10^9. y. Required the product of c^ and a^. Ans. a^i^ or ^^ PROBLEM VII* Xo divide one surd quantity hy another^ ^ RULE. 1. Reduce the surds to die same index. 2. Then take the quotient of the rational quantities, and to it annex the quotient of the surds, and it will give the whole quotient required 5 which fnust l)C* reduced to its post simple terms. E!tAMFLES. 1. It is required to diiride 8^108 by 2v^6. 8r^2Xy'io8^6=:4/i8=;=4-/9X2==4X3V'2=:i2v?:^ ^e quotient required. 3 3 2. It is required to divide 8y'5i 2 by 4%/ 2. I _l^ X z 8-^4—2, and 5 1 2'^-^2 5 =256^=4X4^.5 Therefore 2X4X 4^:;r8X4^?:8y^4, is the quotient re- quired. 3. Let 6y/ioo be divided by 3y/2. Aps.^ ip\/2t 3 3 3 4. Let 4^1000 be divided by 2^4. Ans. iOy/2. 5. Let \\/ttt ^^ divided by -fy/y. 'Arts. 1^/3. 6. Let >L?t4V4'^^^d^l^TH ' AOT.ffJ/3, 7. Let jVii, or -j^^, be divided by -JaX Ans. i?5^^«». ^ PROBLEM VIII. * • Yo kfvohi^ or raise^ surd quantities to any power* RULE. Multiply the index of the quantity by the index of tho power, to w|;^cb it is to be raised; and annex *Ae result to the poweffbf the rational |>artSj» and it will give the powe^ l^equired. * - , EXAMPLES* I. It is required tp find the sq^uare of ^ii ^ Firsts |i*=^XT=i> And Ji*=/^^=,-^=^Ji Therefore ffl^* =?-i-<i*F=?:|Vfl% the square re* quired. ^ * 2. It is required to find the cuhe of \*/ls First, 5-=XX|Xi=^Tifi • 7*1 =.7^=7^1 'Therefore 'Wi^^l^'f^^^W^^'y the euba required. 3. Ret}uired the square of 3V'3. Ans. pvp^ 4. Required the cube of z* » or ^2. Ans. 2^/2. 5. Required |. Requii^.the 4th power of -yV^* Ans. <S. It is required to find the «tii power of a"*. "^ An8,7j^ # ,F&OBi.£M IX« Tc extract the roots of surd quan'tttUls RULE.* Divide the index of the giveii quantity by the index of the root to be extracted ; then annex the result |^ the roo^ of the rational part, and if. wiU giye the root required. l^XAMFXES. 3 1. It is required to find the square root of 9y^3« First, ^9=3 5 , . And 3*1^= 3^ -^^=3^ 5 Therefore ^^^ | ;= 3 X 3^ is the square root required. * TJie square root of a binomial or .'residual surd, ^+B9 or ^r-^ may be found thus : take V-^* — B^=:D 5 2 2 Thus, the square root of 8+2-/ 7= 1 + ^/7 ; And the square root of 3 — \/8=:^2 — t ; Put for the cube, or any higher root, no general rule is giyea, . JL!0^ AtCBB&l. '^ 'f' 9. It it required to find the cnbe root of IVju First, •i=-J-j And ':;77r= 2*-*-33 27 J Thercfarc •J^^^zj'^rsBiXa^ is die cube root required. f 3. .IteqaiTed the square root of lo^ Ans. io\/io. 4* Required the cube root of -^^^^ Ans. \{u 5. Required the 4th root of yi^^ Ans. 3^X^*t» INFINITE SEI^IES, An infinite series is formed from a fraction^ having \ compound denominator, or by extracting the root of a surd quantity ; and is such as, being continued , would rui^ on infinitely, in the manner of some decimal fractions. But by obtaining a few of the first terms, the law of the progression will be manifest, so that the series may be continued without the continuance of the operation, by which the first terms are found. PROBLEM I. ?i reduce fractional quantities to injinite series. RULE. Divide the numerator by the denominator ; and the op-t cration, continued as far as may be thought necessary, will give the series required. EXAMPLES. EXAMPLES. !• Reduce to an infinite scries. X— ;f) I ( i4-A?+;c*+*^+;v*4-, &€•:?= , and is tte I— ;^ answer* i] -H*,-*» +**'* Here it is easy to see hofw the succeeding terms of the quotient may be obtained without any further division. This law of the series being discovered, the series may be ccmtinoed to any required extent by the application of it 2. Reduce --— to an infinite series. -4- = I — x+x'^'^x^^'X^ — > &c the zm^jttt. ' Here the exponent of x also increases continually by r from the secoml term of the quotient i but the 9iga& oi jhc terms, are alternately + and —. 3* :|R.ec|uce flat w ttCSBEA* o. Reduce —7-^ to an infinite series* - ^^ ■" .' ■ * ex ' a ^7 -. +1;+ > • • + -4. &c and is the answer. 4« Reduce * Here we divide c by a^ the first term of the divisor,' and the quotient is — , by which we multiply a+x, the whole ^vi- sor, and the product is — | — or C'\ — , which being subtract- ex fA from the dividend o there remains -— — ; this remainder, be- a cx ing divided by a, the first term of the divisor, gives '^ for the ad term. iN^imTB Sfi&XS^i 33p 4. fteduce — - to an infinite Series* Ans. -J X : 1 + -^. + ^+^. &C. J. ftcdUcc I term of the quoti^t, ty which we dso multiply ^4"^, the divi- acx sor, and the product is ^ p>- oJ^ — — — * « ^ ■ | \vhicb, being taken from — — , leaves -\ — -. The rest of the quotient is found in the same mataiietr j and four tem\s being obtained, as above, the law of Continuation be- comes obvious ; but a few of the first terms of the series are gen- ^lally near enough the truth for most purposes. And in brder to hive a thie series, tbe greatest terto of the dl^ tisor, and of the dividend^ if it tonsi$t of more than one term, inust always stand first. Thus in the lait example ; ii* ^ be greater thaiii a^ then x must, be the first tefhl 6f the divisoir, ihd the quotient will be — — nir Af-f-tf . , c &c a*c a*e . % . — — — T H T" — — 27" +>&c. the true scries ; but if ;c be lea^ than a^ then tiiili series is false, a6d the further it ii» cD&tinaed» Ac more it wiU diverge from the truth. For let d±:2, cni and xz:±i ; then if the division be petforin->' ed with a* as the first term of the divisor, you will hate — r— a 1 , But if * be placed first in the divisor, then will — p- itt | s5 iqp^ =1—24-4— 8+ if —, ^c* Now it i^ obvious, that the first series continually converges to. the truth f for the first term thereof, viz. t* exceeds the truth by. S 8 =i-i5: + i-^V+»&c. =J. 5. ftedoce --— .ip m infinite ftcrieB. ' 4f^ 'I^edttce —7-7* ^ ^ infinite series* Ans.i~^^;;^-;jr. are. ' 7, Rediiee ^ ' to an infinite series. Ji J. 1. PROBLSII j m^ I II I II I ■ I 1 ■ ■ ; i I ill - I . i-— b ^ 7 I ^^ ^^'^^ are deficient by -n i tbiree terms exceed It. by ^ ; foar terms are deficient by ^ ; five terms wUl exceed the truth by -^f 8cc. So that each succeeding term oF the series brings the tjuotient continually nearer and nearer to the truth bf etc half oif its last preceding difference ; and consequently the series will ^)proximate to the truth nearer than any assigned ooia* ber or quantity whateyer ; and it will converge so much the swift- mf as the diifiBor is greater than the dividend. , But the second series perpetually diverges from the truth ; ibr the first term of the quotient exceeds the truth by x-^» or 7 | two terms thereof are deficient by 4 » three terms exceed it by | ; four terms are deficient by '•/ ; five temis exceed the truth by Iff Stc« which shew the absordity of thia series. For the same reason, » must be less than unity in the second example ; if x were there equal to unity, then the quotient woulJ^be alternately t| and nothing, instead of i ; and it is evident, that x is less than tmity in the first example, otherwise the quotient would not have been afirmauve ; for if x be greater than unity, then i — Xf ther" divisor, is negative, and unlike rigns in division give negative qno^ «ents. From the whole of which tt appears, that the greatest term of the divisor muse always stand firsic. . raOBLEM lU To reduce a compound surd to an infinite series. aviE.* Extract th# mot ts io Arithm^!c» and t^ apentiooy ^ontinited as far as^ may b« thoMght necessarji will give tbe series rec}iiiite<U I. Required the square root ota^JfX* in au infinity 'feries* ^9^)H^ ir X* ' »* «»* J!l ill .1 „** 2« Required * llus nile IS diiefly of use in extractbg the square root ; the .fperadon beiog too tcdioast when it is apjdied to the higher JlOWCfk f Here the square root of the fint tenn, «*, u «» the first %EnB tf 4he root, wfaidi» being squared and taken from the grten «9r4 41*4- «(% kav^ n* %%m nmrnuiap dhrided bf a«^ tvioe the 2. Requbed the square root of #*«~a(* in an infinHe feriet. f ). Ooif^ert ^i-f-i inla ui io&nke'Kiiei. ' ' 4|. Ijet ^fh^x* be joonverted iato an infinite series. |. Let ^i—K^ be conrertcd into to infinite series, t. SIMPL?. I the first term of the root, gms *— fbc the second term of th^ foot, whk^i added to aj, gives ta-| — for thd first coo:^ pound d^visoTft wjiick being multipfied by — t aod the produ^t^ x^ u. ...^ taken from the first remainder k*, there remains -^ — r tlus remainder, divided by ^a^ gites •— s"-r for the third term of aOr. . * . ihe root, which most be added to the doubk of a+ — , the two ■ ^' ■ * ■ ?^ ■ ;. first terms of the root, for the next compound divisor. And b^ proceeding th.us,. the series ri^ay be continued as far as is desired. Note. In order to have* a true series, the greatest term of tbe pro|Kxse4 surd nuiit \ft alwyys placed frit 81MPIE EqpATlOMt, 33J SIMPLE EQUATIONS.. An Equation is when two equal quantities, differently expressed, are compare4 togetl^er by means of the sign == placed between them. Thus, 12 — 5=7 is an cquadon, expressing the equality of the quantities 12-t-S and 7. A simpU equathn is that, which contains only one unr ^icnown quantity, in itf siniple form, or not raised to any ppwen Thus, X'-^a-^^bzzzc is a simple equattojiji cqptaining on- ly the unknown quantity ^. Reduction, tf equations is th^ method of finding the valu^e of the unknown quantity. It cbnsists , in ordering the equation so, that the unknown quantity may stand alone fPn one s;de of the equation without a coefficient, andal^ fhe rest, or the known quantities, on the other side. RULE !•* Any quantity may be transposed from one side cf the ^q\i:^tion to the other, by changing its sign. Thus, if ^-4*3=7» ^^^^ '^^^ ^=7"t3=4- And, if X-— 4+6=8, then will ;i;=8-f-4'^— 6zz6. Also, if If— '^+*=^^— ^rthen will *f=i:^>— rf-J-j— 3.' And, in like manner, if 4^— •8^3Ji>+2o, then will zp? ?— 3^z=;2o4-8| or;ifz;:28. ijiyLE * These are founded on the general principle of pcrforipinf ^qual operations on equal .quantities ' when it is evident, that the results must still be equal ; whether by equal additions, or su^ ^raction^, or multiplications, or. divisions, or roots, or powers. D4 aCSMVLAn &ULB at If the imkMirn term be multiplied hj any ffUMStjp Aat quantity may be taken away by dividing aU the othef tenns of the equation by it. Thusj if tfir=pii"'-n0y then wVl (Kssi^^t. And if 2j;-|-45^x(i^ then will x-^azzS^ and ;r^=pS-^a In like manner, if oK-^Tjtass^c* ^ tben ^ill, xJj^iIfSS '^— , and *Ss: — -^2i, RUM 3« If the unknown term be divided by'any quantity, that quantity may be taken away by n^ultiplying all the othe^ termi c^ the equation by it. Thus, \f - =5<>^3, thep wiIlfr::;io4-ti3m?i9» And, if - z;;^ I itj tj , then will A;::^^i^-|-tf<>^-«£t • In like manner, if — -^2=64*49 then will 2x— -ds3i 18+12, and 2^318+124-6=36, or *5;5:V=^^8. EULB 4. The unknown quantity in any equation nuiy be made f!rce from surds by transposing the rest of the terms ac<^ cording to the rule, and then involving each side to such ^ poweri as is denoted by the index of the said surd. Thus, if \/x— ^2=:;6, then will y^;v=6+2=8, ^nd| %=8'=64. And,if v^4X+x6si% then will 4;i;+i6:=5 144, and 4«cri44-^i6=:i28, or xas — =$a. * 4 In SIMPLE EQt7ATf0k$. 33^ In like manner^ if •2^+3+4=8, then will v' 2^4-3 And 2ir*|p3=V=*4f «id 2;ifs6i|r-*3s^iy or *~ V HULB ft If that side of the equation^ which contains di^ tin* known quantity, be a complete power, it may be reduced by extracting the root of the said power from both sides of the equation* ^ Thus, if af*+6;v-|-9=raj, then will *+3=V25tsj, or*f=:S-*-3ssa. And, if 3A?*— 9=5 21+3, then will 3flf*5c2i+3+9« 33, and*'=:y = ii, orxzs^iu InJikc manner, if -f-iosssf, then will 2X*^^0 sr6o, and;i;'*j-i5a»30, orir*«3o— 151* 15,01 «=:i/ij» ♦ W^LE (J. ■ ' Any analogy, or proportion, may be converted into an equation, by making the product of the two mean terms equal to that of the two extremes. Thus, if 3^ : 16 :: 5 : 10, then will 3^X10 22 I6X J, and 30^= 80, or ;vs-|^s2|^ And, if — : a :: f I Cf men will -*— 3?^*, md 2CMZa 3 3 ^ahf or *f=— . In like manner, if 12-— # : — :: 4 -s r, tfaenwill ta-«f 4^ If = — =s 2X| and 2;ir-f >r3£ t^j or-sf s y s 4. RUtS 120 AtaBB&i. EULE 7. If any quantity be found on both sides of the equatioii with the «utae sign^ it may be taken away fron\ them both ; tnd if erery term in an equation be multiplied or divided by the tame quantity^ it may be etrucli: out of them all. Thu^ if 4jr4-tfz^^-tf» then will 4*^=*, and x=^^ J^nd, if ^aic^saizzSatp then 'will 2^-^$k±2c, and 3 2* in like manner, if — — j=:'y — -y, th«i will 2#f2a i5, andxzzS. - MiSCEJULdNMOVS EXJM^LMSl. t. Given 5Af— j^=:2;r+6} to find die value of itf. First, 5* — 2*=6-[-i5 Then 3Af=2i AndAf=i:V=7- a» Given 40—^—16=120—714^ ; to fiiid ^. First, I4X—6ac=I20^— 40+16 Then 8^=96 And, therefore, xtsi^-^ = 1 Zm 3. Let 5fl;c — 3i:=2rfx-|-^ be given ; to find ^\ First, 5/JAf— 2^A;=zc-f-3* Or 5^— 2rf XAf=:f+3* r+3^ And, therefore,.flf=i 2d 4. Let iilCPLE EQiJAXXOMS. $3) 4; Let 3^*— io;f=8x-|-;c* be given j to find^« First, 3ir — lorrS-f-* . - ■ ' And then 3*— i*=r8+io Therefore ax=:i8i and ;r;=:V r^p. 5; Given 6jx''-i-iaa*;c*sr3tfAP'+6^;c* ; to find X ]tirst^ dividing the whole by yx*^ we shall have And then 2x-i-ix=:24'4* Whence *z=2-f-4^* ^. Let — -^— -+■ - =± ^cJ be given ; to firid Ki a 3 4 First, X— — +— = 20 And then 3Af— alf^ =60 And I2X — 8flf-|-^=240 Therefote 10^1=240 And ;c=Vtr=a4. AT— r-3 ^ ^-f'^P A . 7. Given — r^ -4 srao*— — : 1 to find m 2*32 'XX First, X— 3-| - = 40— AT— ip And then yc — 9-f-2Arr=i2o — '3^: — ^ And therefore 3>f -}- 2^+3^:^= 1 20 — j y-j*"?^ That is, 8x-=72, ovx^::^-^—^ %. Let ^ja:-|-5=7 be given ; tp find ,v. First, ^jX=:^ — 5=2 And then jXZZ 2 * =14 And 2a;=;:I2, or x:=^ ^p* ZI(J. ^ y 9' ifii -y •'«««»«M-*-'''<^ '■'■•■ "/ Vi' \ '"tii"* 9. IiCt^v/fl*-|-x* sr '. ^Uip SS: b^ given; to find jip. •First, irv^^^-H^^f^^-f***— 2^* And tiien>v^df«^«i±^«'=.J-«r' And i^^X^=^**=^*— *1*— «*— aaV+v* Whence /i*^»+afl*i*=a^ Or 3a*«*=ai* /I* And consequently j(f* — ■' ■ j 3a . /I* And X=v/ — r =^\/t- 3« 'J £xaaSplms fOR Practice. U Given x^^^iSzz^x — 5 ; to find-^. Ans. /irrrui. 2. Given 3j^— fl+^z^rrrfj to find j^. Ans- ;izr ■ 3. Given ^— ^/i?+i0=:2O— 3l^^2 5 to find «'. Ans. xrr2. 4. Given 3/1* ^-"^ — •3=:iA^-^tf ; to find x. Ans. xz 6flr— 2^* XXX 5. Given — + -r -4 =:^ > to find ;tf. • ^ * ; ^ ^^ Ans. xz fc+^jr-|-fl3* <S. Given — -4 — — —=4. ; to find ^v. Ans. xzzzL a ■ 3 4 7. Given v/i24^=:2+v^;f ; to find x. Ans. *=4. 8. Given SIMPtC EQPATIOKS. 331 'X ^ . z '^ 8. Given \/a'4-x'=b^'f^9c'^\l torffnd^r. 2a- 9. Given x-f-<»=v^a*-l-^V^**+^* ' ^° ^'^^ ^• JIEDUCTION OE TWO, thUeE|, q:el mqb£, Simple E^ATIONSj CONTAimKG TWO, THR^E, OR MORE, Unknown Quantities, - PROaiaEM I*^ ,.:_v ^: ^ 3^(? exterminate twa unknown quaniitiesy or to redtice the twa simple equations CQttfgioing thetn to-one* . .. ...'-. : i^ULE I. , ?■ ' •" ; I. Observe which of the unTcnown quantjries is the least involved, and find its value in each of the equations, by the methods already explained. - 2* Let the two values thus found be made equal to each other, and there will arise a new equation with only , one unknown quantity in itj whose value may be found as before. EXAMPLES. I. Given < /T'3)' ^ 3 C . to ?ind\v and j. ISx — 2;=io 3 ' ^ From the first equation Xz:=z-' — ^> 1 0*4- '2 V And from the second iv=zz , ' ■ • "5 And consequentlv •— 23— ^_io+2v 5 X |49 4tCK»a4. Or M5-riS;=20-jr4)>, Or xg)=i 15—20=95, Wheacc kzz =4- s* Given f:is|-. to find ;ff and jt« From the first equation «(=zs2i---^, And from the second, ^rri+J'* iTxcrcfore i?— 5'=::*-j-y, or 2j^i=flf— 5, And consequently, y=^ , And yzr a — j i .y-j g"— ^__2+5 • 'j. Glyen -^ 2 3 4 + f==8 N ; to find PC and j^ From the first equation Af=i4 ^, ■ • ^ ' 3' And from the second, ^=24— — , 2V ^V Therefore, x 4 '" =24 — ^, ■3 • •• 2- An4 42— 2;;=:72— -, Or 84 — 4;= 144 — 9y'* ■ Whence 5;? 3: 1/^—78 4 =60/ And;=:V=^^' 2j; And xz=z\^ =14 — V =^* 3 ^ ^ - 4. Given IIMPLE EQUATIONS. 34;!, 4* Given 4x4-jz:34, and 4^-1^^16,5 to find x and j. .' ; Ans. Arri8^ aod^z^ra. c. Given — +— =-^, and — +'-^=^'rr^ 5 to find 5 4 ^ ^ ■ 4 . JJ tS and jy. ' Ans. ^==x* and^-zzr-^ 6. Given ^r-J-j^rr/, and x*» — y^:rzd ; to find x and jr. . Ans. 4^=; "! * ; and ^:;z r- 2J ^ "^ 2J .Ru;-p: a. 1. Consider which of thejunkaown quantities you would f rst exterminate, and let its value be foupd in tha|: equa- tion, where* it is least involved. 2. Substitute the value thus found for Its equal in the other equation, and there will arise a new equation \vith only one unknown quantity, whose yalue mfiy be found a§ before. EXAMPLE?. u Given \ ^"T^)'— ^7 I . ^^ gnd at apd 5. ' IZX— y= 23' ^ ^ From the first equation ;v= 1 7 — 2j», And this value, substituted for x in the, second, gives 17 — 2yXZ'r^y=^9 Or 5ir— 6>T->'=2, or 51— .7>'=r2 ; That is, 7j;=:5 1-^—2=49 > ^ Whence ;>=y=7, and ^=17— *2/=i7 — I4rr3, »2. Given \ f^ ^ > J to find at and y. From the first equation xzs. 13— y. And this valup, being substituted for x in the second, Givesi3— J— y=3, or 13— 2j^=3J That is, 2j>=i3— -3= 10, ' , Or y='-^z=:S, ,and «r=i3— 7=i3r-5=r8. 3. Give^ % The first walogy turned. iQtQ an e^itacioa. Ib ^x=::;d!;9 or Jifs* -j, At}d this nliie of ;rj substjt^'ted ia tlicsecQ^^ji Or aY^'y'^'^** or/'c J^, I—it: And *ewfow, >=^;q7.| » *°'* «= ;iaj:| ' 4* Given — +7J'=^98i ^^^4 T +7*^=^5 1>. to fin4* »nd j^ An& >=;; 7» and fszi^^ X If ' SC"^^ X c. Given — -^-laae — +8, and — ^ 4- » 8 =» *^. 2 • - .4 •■ 5 .3 - - ■* }-27 i to find A' andj^.. ' Ans. ;c5?6o, and j?— 40^ 4 6. Given fl \ b m x \ y^zaix^- — y^zzf/; tofind/r and;?* _ ^1 » RULE 3. Let tlie given equations be multiplied or divided by such numbers or quantities, as will make the term, which con- tains one of the unknown quantitiejS, to be the safme in both equations } and then by adding or, SJubtractixig the equations^ according as is required^ thei^^ will arise a new equation with only one unknown quantiityj^ as before. SXMPLfc EQJTAsriONS. \ ^ ,3^3 EXAMPLES, 1. Given ? 3^+5)' = 40 ? .^ fij^j ^ 3j,^ First, multiply the second equation by 3, * And we shall have 3^+6^=42 ; Then, from this last equation subtract the first. And it will give 6)^—5; =1:4 2— 40, or ^ =: 2, ^ And therefore, ;>;= 14— 2jr'32 14—4= 10. 2. Given s ^ I ^-^"^ ^ r ; to find at and v. i 2xhJ-5j;= 163 Let the first equ^tioti be multiplied by 2^ and the $ec- oiidbyr.5. And we shall have io;c— 6y= 18 And If the former of these be subtracted from the latter, It will give 3137=262, or j?=-|-fzi:2, o«4-3v And consequently, jvn: , by the firit equation, Another Method. Multiply the IJrst equation by* 5, and the, second by 3, And we shall have 25^1:— 4537 = 45 IJow, let these equations be added together, And it will give 3ia: = 93, or a: = |4==3» And consequently, y rs , by the second e^uatlonf 16— .6 Or v=: ■■' =; V = 2, as before- MlSOSZLANZOVS 344 ^GEBRifr. Miscellaneous Examples. 1. Given l-8v3=3i, and --^ X-ioiCzz igi ; W 3 4 find ;if and ;■• Ans. x'zzi^ and j = 3. 2. Given 4-14:^18/ ind ■ f-ifizrio ; to a, 3 find a: and jt Ans. ^^3=5 and j; =2 2. 3. Given — - — • -I =8, and -^ ^ ^z=. ii 5 td •> 6^3' 2 -^ -' find X and ^'. Ans. x=:6 and j;=±8^ 4. Given aX-\-bj^C9 ^^^^ dx-^-eyzzf'^ to find /i: and j^; A ^^ — y , ^/^^ Ans. x= rr andicr -^ PROBLEM II. JTi txterminaU three unhnown quantitieSy cr to reduce the threi iimple equations containirtg them to onei HULE. , i. Let iCi y and z, be the three unknown quantities to ' be exterminated. 2. Find the value of x from each of the three given cguations. 3. Compare the first value of x with the second, and an equation will arise involving only y and 2. 4. In like manner, compare the first value of x with the third, and another equation will arise involving only y and z. 5^ Find the values of y and z from these two equatibns, according to the former rules, and x, y and z will be ex- terminated as required. Note. Any number of unknown quantities may be ex- terminated in nearly the same manner, but there are offcn much SIMPLE EQUATIONS. 34S mnth shorter methods for perfotming the operation^ which will be best learnt from practice. X. Given EXAMPLES. ^+ y + 32=62 — I — 1-— =10 ^i to Gndx^ymdz.^ From the first equation /vrrap— -j?— 2. From the second ;fz=:62— -2^—^32. 2« „ 2 From the third af=:2o— -*>*-—. Whence 29 ■ j > ■ ■ 2 ^:62—2^—32, 2y z . And 20— -v— r2=:20— ^ — p— ; 32 ' But, from the first of these equations, j'^da— 29— 2» P33— ^j 32 ' And from the second j?ir27— • *"" 5 Therefore 33— 22=27 , 0T2=:::i2a And ^'1=62 — 29 — 222=62—29—24=9, And ;vzz29-*— ;; — ^2=29 — 12—9=8. 2. Given < X y z — -4- — -i- "" 1:^62 2-34 j j =47 ^5 to findA',jiand«t Fust, V V 34^ ILGBBRif . First, the given equations, cleared o£ fraeticmsi I2;c+ 8H" 62=1488 Then, if the second of these equations be subtracted from double the first, and three times the third from fire times the second, we shall have io*-f-3;'=42o And again, if the second of these be subtracted from three times the first, it will give 12^ — 10^=468 — 420, or Af=*-5? =24 ; Therefore " y=z 1 5 6-^4^=6, ^ , 1488 — 8v — I2Ar And s= ■■ ■ ■ 7 mao. 6 3. Given *r+v-|-s=3i, x-^y — 2=25, and 9 C y z iz^ \ to find jv, y and z. Ans. a;=2o, jc=8 andzz=:3* 4. Given .v+;iZtf, A;-j-2=3, and ;^-|-2nr ; to find x^ ^ and z. 5. Given < rf;f^-^j^-4:^=='' r > to find ^, y and «. A Collection of ^UESTlONSy producing Sim-^ PbB E^ATIONS. I. To find two such numbers, that their sum shall be 40, and their difference 16. ■ Let X denote the less of the two numbers required, Then will Af+i6==: the greater, And ;f-|-^"H^6=4<^ ^7 ^^^ question ; That is, 2X=r4o — 16=: 24, . Or x:=i*-^ zziizz less number. And x-|-ifc=:i2-^i6=28= greater number rcquired- 2. What SIMPLE E<^UTJ0N3. 347 2. What number is that, whose ^ part exceeds its -J part by 16 ? Let A? equal number required. Then will its y part be — , and its -^ part •— ; X X And therefore - ^— — =s=i6 by the question, That is, ^— — =48, or 4^;— 3Ar=: 192 ; 4 Whence xzzigz the number required. 3. Divide loool. between A, B and C, so that A shall have 72L more than B, and C lool. more than A. Let ;i;nB's share of the given sum, Then will x-|-72=A's share. And X'\- 1 72=:Cs share. And the sum of all their shares x-]-Af-|-^2-j-;v-|-i72, Or 3;if+244ZZiooo by the question ^j That is, 3X=: 1 000— 244=756, Or /c=:^-j^r:252l. :^=z B's ahare. And ^-|-72=:252-f-72=324i; =±::;A*is share. And /c-f- 172=252+ 172=4241. = Cs share. 25 2I. . 324h I cool, the proof. ^ 4. A prize of loool. is to be divided between two per- sons, whose shares therein are in the proportion of *j t» 9 ', required the share *of each. Let X equal first persons sfaisire. Then will 1000— ;v equal Wond person's share, And X : looo — x :; 7 : 9, by the question, Thjit 348 , AL6EBR1. That Is, 9;i:=:iooc — xX 7=7000— 7^ Or 16*^:7000, Whence xzr — ^-=437!. los. zz first share^ And 1000 — *=iooo— 437L 10$. zs 56^1. los. 2d share, y. The pav5ng of a square zX 2S. a prd cost as much as the inclosing of it at 5s. a yard ; required the side of the square. IjCt X equal side of the square sought. Then /\x=: yards of inclosure. And .V * 3; yards of pavement ; Whence ^xXs^'^.ox equal price of inclosing, • And ^*X2:ir2Ar* equal price of paving. But 2X* = 2oxby the question, Therefore, x^::ziox, and ;v== 10 = length of the sW^ required. 6, A labourer engaged to serve for 40 days upon these conditions, that for every day he worked he should receive 2od. but for every day he played, or was absent, he was to forfeit 8d. ; now at the end of the time he had to re- ceive il. IIS. 8d. The question is to find l^ow many days he worked, and how many he was idle. Let X be the number of days he worked. Then will 4c — x be the number of days he was idle ; Also yX^o=2 2ox=: sum earned. And 40-^x X 8 = 3 20 — 8^ = sum forfeited. Whence lox — 320 — 8Ar=38od. (zzil. lis. 8d.) by t^e ijucstion, that is, 2qa: — 32o-j-8Af=38o, Or 28a: = 3 80+320 = 700, And xnz'^-^ = 2511: number of days he worked. And 40— ^»v:;= 40-^25= 15=; number of days he was idle. 7. Out SIMPLE EQUATIONS. , 349 7. Out of a cask of wine, which had leaked away y, Zi gallon's were drawn ; and then, being gauged, it ap- oeared to be half full : how much did it hold ? Let it be supposed to have held at gallpns. Then it would have leaked — gallons, 3 And consequently there had been taken away 21-] gal. But 2i-| zz — , by the question, ^ That is, 63+Ariz: — , . Or i26H-2;v = 3Ar 5 Hence 2^ — 2.v=i:^6, Ox x:=i 1 26::;= number of gallons required. 8. What fraction \s. that, to the numerator of which if I t)e added, the value will be y ; but if i be added to tha denominator, its value will be ~ ? Let the fraction be represented by -»•, ' . y Then will — = -f-. Or 3^+3 =y^ And 4^iii)-|-i, Hence /\x — 3^ — 3iz:j-j-i--y, That is, X — 3 ?= I, Or:ifiz:4, and y= 3^4.3= i2-f-3= 15 ; So that -j^ zn fraction required. 9. A market woman bought a certain number of eggs, »t 2 a penny, and as many at three a penny, and sold them all again at the rate of 5 for ad. and, by so doing, lost 4d. What number of eggs had sl)e i Let 359 ALGEBKA. Let ATS number of eggs of each son* X Then will — =: price of the first sort. And — zz: price of the second sort. AX But 5 : 2 : :• 2* (the whole number of eggs) : — \. Therefore — price of both sorts togethetf at 5. for 2d. X X AX And {-• — -— — = 4 by the question } That IS, xA = 8 ; 3 5 24X Or 3^+2^ 7- =24, 5 ^ Or 1 5x4-1 ex — 24^= 120 ; Whence a:=:i2o= number of eggs of each sort required. 10. A can do a piece of work alone in ten days, and B in thirteen ; if both be set about it together, in what time will it be finished ? Let the time sought be denoted by x, X Then 1 o days : i work : : x days : — , X And 13 days : i work ; : x days : — , 13 X Hence — = part don*^ bv A in a: days ; 10 . ' And — =z part done by B in ^ days. X X Consequently, \ n i ; 10 13 SIMPLE EQPATIONS. 351 That IS, —-}-*= J3, or i3^-|-io;f =: 130 ; And therefore 23^?= 130, or ;tf = \^ = 5|^ days, the time required. 1 1. If one a^ent A alone can produce an effect e In the time ay and another agent B alone in the time b ; in what time will they both together produce the same effect ? Let the time sought be denoted by x. ex Then a : e i\ x \ — =: part of the effect produced by A, And * : f : : ^ : -- = part of the effect produced by B, Whence h T=^ hy the question ; b , ', . .ax That is, AT+Y^^ » b Or bx^^xzizba ; ba And consequently, *=7^j--=: time required. ^ESriONS FOS PRACTICE. 1. What two numbers are those, whose difference is 7, and sum 33 ? Ans. 13 and 20. 2. To divide the number 75 into two such parts, that three times the greater may exceed seven times the less by 15. Ans. 54 and 21. 3. In a mixture of wine and cider, ~ of the whole pluf» 25 gallons was wine, ~ part minus .5 gallons was cider ; how many gallons wer^ thei^e bf each ? Ans. 85 of wine and 35 of cider. 4- A 35 a ALGEBRA. 4. A bill of 1 20I. was paid in guineas and moidores, and the number of pieces of both sorts used was just 100 ; bow many were there of each ? Aus. 50 jof each* 5. Tw^o travellers set out at the same time from London and York, whose distance is 150 miles j one of them goes 8 miles a day, and the other 7 5 in what time will they meet i Ans. 10 days. 6. At a certain election 37^ persons voted, and the can- didate chosen had a majority of 91 j how many voted for each ? Ans. 233 for one and 142 for the other. 7. There is a fish, whose fail weighs plb. his head weighs as mudi as his tail and half his body, and his body weighs as much as his head and his tail ; what is the • whole weight of the fish ? Ans. 721b. 8. What number is that, from which, if 5 be subtracted, ■5- of the remainder will be 40 ? Ans. 65. 9. A post is one fourth in the mud, one third in the wa- ter, and 10 feet above the water j what is its whole length ? Ans. 24 feet. 10. After paying away one fourth and one fifth of my money, I found 66 guineas left in my bag j what was in it at first ? Ans.' 120 guineas. 11. A's age is double that of B, and B's is triple that of C, and the sum of all their ages is 140 ; what is the age of each ? Ans. A's = 84, B*s =42 and Cs = 14. 12. Two persons, A and B, lay out equal sums of money in trade ; A gains 126I. and B loses 87!. and A's money is now double that of B ; what did each lay out ? Ans. 30CI. 13. A person bought a chaise, horse and harness, for ^ol. J the horse came to twice the price of the harness, •and -^ toffPLfi fiQJTATiOKS. 35 j and the chaise to twice the price of the horse dnd the har« hess ; what did he give for each ? Ans. 13L 6s. 8d. for the hbrse, 61. 13s. 4d. for the har-i xiess^ and 40I. for the chaise. 14. Two persons, A and B, have both the same in- icome ; A saves one fifth of his yearly, but B, by spend- ing 50I. per annum more than A, at the end of 4 years finds himself lool. in debt 5 what is their income ? Ans. 125I. 15. A gentleman has two horses, and a saddle worth jjfol. Now if the saddle be put on the back of the first horse, it will make his value double that of the. second ; but if it be put on the back of the second, it will make his value triple that of the first ; what is the valiie of each horse ? Ans. One 30L and the other 4SJL 16. To divide the nuniber 36 into three such parts, that ^ of the first, y of the second, and -^ of the third, may be all equal to each other. Ans. The parts are JS, 12 and 16. 17. A footman agreed to serve his master for 81. a year ^and a livery, but was turned away at the end of 7 months, and received only 2I. 13s. 4d. and his livery 5 what was its value ? Ans. 4!. 163^ 18. A gentleman was desirous of giving 3di a piece to some poor beggars, but found, that he had not money enough in his pocket by 8d. ; he therefore gave them each 2d. and had then 3d. remaining ; required the nu<nber of beggars* Ans. 11. 19. A hare is 50 leaps before a grey hound, and takes 4 leaps to the grey hound's 3 ; but 2 of the grey hound's leaps are as much as 3 of the hare's ; how many leaps must the grey hound take to catch the hare ? Ans. 300. 20. A person at play lost -J of his money, and then won 3 shillings ; after which he lost j of what he then had, and W w 354 ALCSMXA^ and then won 2 sliillings ; lastly, lie lost y of what he tlien had, and, tlus done, found he had but 12s. rem^- ing 'f what had he at first ? Ans. 20s. 21. To divide the number 90 into 4 such parts, that i£ the first be increased by 2, the second diminished by 2^ the third multiplied by 2, and the fourth divided by 2^ ' the sum, difierence, product, and quotient| shall be all equal to each other. Ans. The parts are 18, 22, 10, and 40, rcspectivdf^ ' 22. The hour and minute hands of a clock are exactly together at 12 o'clock ; when are they next together ? Ans. I hour, 5-j^ mm* 23. There is an island 73 miles in circumference, and 3 footmen all start together to travel the same way about it 5 A goes 5 miles a day, B 8,. and C 10 ; when will they a!l come together again ? Ans. 73 days. 24. If A can do a piece of work alone in 10 days, and A and B together in 7 daysj in what time can B do it alone I Ans. 23 J days. 25. If three agents. A, B and C, can produce the ef- fects a, ^, Cf in the times e,/, gf respectively j in what time, would they jointly produce the effect d ? Ans. '^^ trme. 26. If A and B together can perform a piece of work in 8 days, A and C together in 9 days, and B and C in 10 days ; how many days will it take each person to per- form the same work alone ? Ans. A 14-34 days, B i7^f, and C 2^-^. . QUADRATIC EQUATIONS. A SIMPLE QUADRATIC EQ^jATiON IS that, which involves tlie square of the unknown quantity oply. An jQJTADRAtiC EqjJATIONS. 355 An AtrccTEn (^adratic eqjjation is that, i«^hich in- Tolvcs the square of the unknown quantity^ together with the product, that arises from multiplying it by some known quantity. ^> ^ Thus, ax*sLHsz simple quadratic equation, And ax*'\'bx:s:c is an affected quadratic equation. The rjxle fox ?> simple quadratic equation has been given already. All affecte^d quadratic equations fall under the three following forms. 1. a:*+jx=* 2. X* — aX:=ib 3. X* — aXz=L — b. . The rule for finding the yalue of x, iu each of these equations, is as follows ; J* Transpose all the terms, that involve the unknowa quantity, to one side of the equation, and the kilDwn terms to the pther §ide, and let them be ranged according to their dimensiops* 2. When * Thfe square root of any quantity may be either -f" or — , and therefore all quadratic equations admit of two solutions. Thue, the square root of +*** *s +fli or — ^ 5 for either +«X -}•«, or **-«X — n is equal to +«*. So in the first form, where x -^ ^ h found ^za/ h'{' — ,the root maybe either -|-v^ ^+ ~ * « ■ 4 4 or — v' ^+ , sbce either pf thera being multrplicd by itself 4 ' will produce ^+ — . And thisambsiguity is expressed by writing the 4 uncertain sign + before y^ 3+ — ; thus ^aciv^ i+ — • 4 42 In 1 35< j|tCE9it4. 2. When the square of the unknown quantity has ^ny coefficient prefixed to it, let ^1 the rest of the terms be divided by that coeiHcient. <. 3. Ad4 III the first form, where J«= + \/ b+ f- — Jl, the first value 4 2 of Xf laz. ATsr+iv/ ^+ — — £- is always affirpatiye ; £ot 42 $ince 1-^ is greater than — , the greatest square must necessa* 4 4 tily have the greatest square root ; y' *+ -- — -^ will, therei 4 * fore, always be greater than y' —> or its equal — | and consC;^ 4 2 ' a* n qucntly -|-a/ i-(- — — -^ will always be afErmative^ ' 2 2 The second value, viz. x= — \/ ^+ f ~, will always 42 be negative, because it is composed of two negative terms^ Therefore, when x * -f-^rxzi^, we shall have x=: + -v^ ^4- .^ -r^ 4 — for the affirmative value of k, and xzz — \f ^-f- — — JS for the negative value of x. In the second form, v/here :v'=r + y'^-j- — + — ,thefirst 4 2 ■ ^ value, viz. Arrz-J-^ ^+ — 4- -f. is always affirmative, since 42 it is composed of two affirmative terms. The second value, viz. :<^=: — V^ ^+ — + ~ , will always ■ be negative ; for since 42 QUADRATIC EQUATIONS. JJJ 3. Add the square of half the coefficient of the second \ttm to both fides of the equation, and that side, which involves the unknown quantity, will then be ^ cpmpletc square. .4^ Extract ^+ _ is greater than _, the square root of 3+ _ (*/*+_) , 4. 4 : *. - 4 4 will be greates tlian v'"^* ^^ ^'s e(jual — j and cpiisec[uentJy ^^\/^+— + — is always. ft negative quantity. ThctcforC| 42 ^vhen X* — ^x=:^, we sjiall have ;c=: + ^ ^+ _ -f. —for the 4 * affirmative value of ;jf, and %zi — »/ h'\ -j- — for the tit^ ^vc value of ^, ^,. ^» ^ In the third for;n| vhef? Jf??V^ ^H ^> both the valuer 4 • * •|jf jf Y^iU be positive, jupposmg — is greater than L For the first value, vi?f ^VSS + V^^^ -*+ ""> ^^ evidently afiTrmative, 4 • 2 b^ing composed of two affirmative terms. ' The second value, ■ I 2 viz. y=: — 1/ i + — , is also affirmative ; for since -^ is 4 ■ ? 4 greater tti^ — — f, therefore V"*" ^^ "^ ^s greater than r 4 42 \/ Bf and consequently — y^ — — 3 + -^ will always be 4 4 * $ti affirmative quamity. ■ Therefore, when ic^^^axzzf^iy we shall bav9 35^ ALGEBRA* 4. Eitract the square root from both sides of the equa# tion, and the value of the uuknown quantity w41 be dev lermincd, as required* . Note. i. The square root of one side of the equation is always equal to the unknown quantity, with half the co^ ciHcient of the &ccoud lerrn suhjoiiied to it. Note 2. -All equations, wherein there are two termt involving the unknown quantity, and the index of one is i*ust double that of the other, are solved like quadratic; >y completing the square. n Thus, Ar*-{-.'7^*~/% or ^"-j-/?x*^^, arc the €amc as quadratics, and the value of the unknown quantity may J^ determined accordingly. EXAMPLES. kave xzZf^Y^ — -i + - for the affirmative v^uc of x^ a^ — -v/ — » — B -J- — for the mrrative value of .r. 4 * Sut in tills third form, if ^ be greater than — , ^e solution of 4 the proposed question will be impossible. For, since the square of any quantity (whether that quantity be aflirraative or negative) is always affirmative, the square iQOt of a negative quantity is im- ^* possible, and cannot be assigned. But if If be greater tlian— , 4 then i is a negative quantity ; and consequently y'.^ — ^ 4 * 4 is impossible, or only imagipary, when — is less than b ; and 4 therefore in that case xzz — + ^ — — - ^ is also iroixjssible or * 4 iivaginary. 1. Girm »*+4^fc:i4<5 ; to find ic. First, A?**^4;^-{-4=i4o-|-4±=r44 by completing the square 5 Them ^^x* -J-4X + 4±ry/i44 by. exfracthig the root > Or /r-|-2=ri2, And therefore x'±z 12— i='iOw , 2. Given X* — 6x+8= 8b ; jto find x. First, ^* — -6;if=8o— 8 = 72 by transposition ; Then x * — 6^+9 =: 72-f-9= 8 1 by completing the square ; And X — ^3cz:v/8i=r9 by extracting the ro6t ; Therefore ;«sz9-{-3=i:i2^ 3. Given 2a?*-[-8x — 16=70 •, to find x. First, 2;c*-|-8;c=:7o-|-2o=:90 by transposition. Then Af *4"4*'^=45 by dividing by 2, And ^*4"4^+4=49 by completing the square ; Whence^"4-2 = v^49=r7 by exttactifig thfcrOots And consequently ^inr-^'— 2=5, — * ^^ 4- Given ^x* — iX'^6=:Sy ; to find X. ^X Here ^*— jf-f-2=:i-| by dividing by 3, And ^*— a:=3I-|.— 2 by transposition ; Also x^ — ^-J^— :il— 24-i=3V ^Y completlrtg th<J square. And * — 4-=\/iV=T ^y evolution ;' Therefore x=^ + ^-zzf ^* ^ 5. Given — — — 4-2c4=:424 ; to find x'. Here -— »— — =42 j —■20^=22- by transposition. 2X And AT*- ii =44y by dividing by |. ; 2^ Whence /»f^X (.i=:44|.4*I.=44^ by.complet- 7 3. Jngfhe^uan^ And \ And ^— |=v/44i=<5f^ "^ Therefore xz=6\ + y=7. 6. Giycnax*'^xz=:ci to find ;>;. ' First; ** + — ^=r — by division % a a h h^ c b* ^ Then ^' -f- ~ *"i" — T ^=~ +"^^ '^y completing the square \ 2a A 4^ , 4^1 by cvoi lution ; Therefore K=+x/ —-^ \—. —^ 4« \2a J. Given flAf *— i^-J'^^^ 5 *^ find Af. Hca^fl*'— *x=i— <: by traiispositiooi iUp« -TT — ^== by division ; '•■'•ifz a a \^ Also »* — •"* + -^ == 1^ '— r ^7 completing the square ; ^ ' • ■- And ;v s:+ v^ 1 by evolution % 2a "^ a 4^ ' Therefore a?=:— + v/ h — ;. 2a — a 4a' . 8. Given A?^-)-2a^*i:^ ; to find x. Here, Af*+2^^*+^*=^+^* by completing the square. And x*'\-az=z^h'\-a* by evolution ^ Whence x^zzr^h-^-a — a, \ And consequently x::Zy^ : v/iHH> *^^* 5^ Giveit Q^iDRATIC EQpATXONi. 3(1. 9. Given tfx"—i/if *—<:=— i ; to find x. First, ax"*^bx^=u:'^^ by transposition, And /v**— — flf*=:— - by division ; Also a:''— 7** + ~» = -7- + ^» by completing tht square, And ;v* — — = -^ 1 r by evolutipn 5 Therefore ;.^=r- + v^~ + ^, 2a — ^ a 4/1* And consequently a?:=— +\/^ 1 ^ r . £XAMFL£S FOR PRACTICJU 1. Gi^cn ;v*— 8;^-|-io=i9 ; to find 9C. Ans. ^=9. 2. Given x* — ^x— 40=1170 ; to find x. Ans. xzzi^. 3. Given 3;tf**^2X — 9:2=76 j to find ^•. ., Ans. ^=1:5. x^ X 4. Given — — — +74" =120 ; to find x^ 2 3 Ans. A?=5*4093, &c 5. Given fl;*-[-A?z:j ; to find ;tf. Ans. XZZ^a^^'-rt^ 4' Given x^-^ax-^^izzic ; to find x. 7 T"^ ^ Ans. xz:w^^+i+— ^— — . ^ * 4 a. X X 7. Qiven 3ft2 ALGEBRi. 7. Given *'— j^=:— 4 ; to find X. a* a Ans. ;c=+v/ — — ^i H • ^ _,. /I.V* cx , e g 8. Given "T" — "T + "r — T > ^° ""^ ^* 9. Given 2*v*— ;c*-[- 1 04=7600 ; to find ;v. Ans. ;c=4. 10^ Given 3^^— 2;v-— — = — ; to find x. ■ ^ 9 9 Ans.x=±v/^^^ + T|"' QUESTIONS PRODUCING Quadratic E^ations. 1. To find two numbers, whose difference is 8 and product 240. Let X equal the less number, Then will ;c-|-8 equal the greater. And ;^Xy"|"8^^''*^* + ^— 24° ^Y ^^^ question ; Whence ;c*-{-8A*4-i6r=24C-j-i6z=:256 by completing the square ; Also ;if-j-4 2=^/256= 16 by evolution ; And therefore x=:i6 — 4=112=: less number, and i2-}-8 zz 20ZZ greater. 2. To divide the number 60 into two such parts, that their product may be 864. Let X =: greater part. Then qUADKAgriC EC^IATIONSJ. ^i^ Then will 60 — xzz: less, And xX6o — X'':=z6ox — x*:z:S6/\ by the question, That is, X* — 6ox= — 864 j Whence x^ — 6oiv-f-9oo;=— S64-|-90c:;:^36 by coipplct- ing the square ; Also X — 3o;^:i^36=z:6 by extracting thp root } And therefore ;v=6+3 0=^:3x5= greater part, And 60— x=:6 — 36=124;=: less. 3. Given the sum of two numbers =io(^/), ancj the sum of their squares =5^ {b) 5 to find those numbers. Let xzz greater of those numbers, Then will a — x=^ less j And X*+a — x\ z;z2X^-^a* — 2axzzb by the question, ^ . ^* ^ Or Af'-4- — *-^ax;;z=: — by division, ^ ' 2 2 ' Or X-''^aX-= — -— » — zz by transposition »; ^J* b a* a* ^b'-^a* Whence x^ — ax A zz: -4 :;z= by com-- '4 2*4 4 pleting the square. Also X— — = y/ by extractmg the root \ And therefore ^if^^+y/ ■■■ -|- — rz greater number, ^ , a .2b-^a* — p.b — a a And a +^ = -J, ^ -^ =x: les$. 24 42 Hence these two theorems, being put into numbers, give 7 and 3 for the numbers required. 4. Sold a piece of cloth for 24L and gained as much per cent, as, the cloth cost mc ; what was ^c price of the cloth ? 364 AtG&BE4* Let 9C = pounds the cloth C08t» Then 24— x = the whole gain ; But. 1 00 I K :: X : 24 — X hj the question. Or ;if* =r 100 X24— ">^= 2400—100^ ; That is, «?-{-ioo;if=24oo ; Whence ^*+'ooJif4-256os= 2400+2500=4900 by com- pleting the square. And ^+50=1^4900== 7p by extracting the roots. Consequently, x=:7o— -5 = 20:= price of tlie cloth. 5. A person bought a number of oxen for Sol. and if he had bought 4 more for the same money, he would have paid il. less for each ; how many did he buy ? Suppose he bought x oxen, 80 'J'hen — = price of each, X 80 . . - And -T — z^ price of each, Jf x4-4 liad cost 80L X+:\ *^ 80 80 _ , , But — =z ' " --Jri by the question, X ^ T 4 '' Or 80=— I l-flf, *+4 Or 8c;v+3^o = 8ox+Ar*+4ijf, That is, X* +4x=z:32o ; "iVhence Af'+4;v+4=32c-f-4=:324 by completing the square ; And ^+2zz:y/324=:i8 by evolution ; Consequently xz=:iZ — 2i:;i:i6 =:; number of oxen re- quired. 6. What two numbers are those, whose sum, product and difference of their squares are all equal to each other ? Let X == greater number. And y = less ^ Then And }zz — j—zzxr^9 or s(z::jr^i from th© sccopd f^quation. ' .■ ■ . " * ' Also y^i \y >'+iXj from the first equationj^ Th^is, ;?*—;'= I, Whence Jl'*-— >-j--^=i^ by completing the square ; i^lso jh---|-iry/i^==:v^^= — by evolution ; Consequently yz=:^^ + 1^^^ » And xzzzy^i::^^^^^. 7. There arc four i^umbers in arithmetical progression, whereof the product of the two extremes is 45J, and that of the means 77 ; what are the numbers ? Let X r= less extreme, And y =5 corhmon difference 5 Then Xf x-{^yi x-\z2yi x-^^y will be the four numbers, And S^X^- f3J^=y *+3^J^=45 ? b/the ix'\^yXx'^2y^x*'\'2^y'^v''=n y^^^^^"* "Whence 2^*^=77— 45=32, ^nd y*zz^-^zz:i6 by suj)- traction and division, Or jc=5V^ 161=4 by evolution, Therefore Ar*4"3*!y=^^*+i2^9fzr45 by the first equation. Also ^*-|-i2flr-4-36=45+36ir6i by completing the square, And x-\^:^ \/ B iz=z^ by the extracting of roots, Consequenriy ifrzzip— 6=r3, And the numbers are 3, 7, 11 and 15. 8. To 1 8. To find three numbers in geomelrical progressioxif who$c sum shall be I4» and the sum of their squares 84. Let Xy y and z be the numbers sought } Then xzzziy^ by the nature of proportion^ And ^!Jtfy!^»=845 I'y *e q^<^«tio«^ But x-^-zizzi/^^^ by the second equation, And flp*-j-2^z+z'=i9^5— 25y-f-v* by sfluaring bo A sides. Or :c*-|-2*-^2^* =196— 28;-|^* by putting 2y* foy its equal 2Xz ; That is, iv *-|-2 * '{-y* = 196-r— 28j^ by subtraction. Or 19^5 — 28;'= 84 by equality \ 190 — 84 Hence ;'=: jj — z^:^ by transposition and division. 16 Again, sKuzizy* zzi6^ or x:^ — by the first equation^ z And ^-|"J"H^= l"4"H^*' ^4 l^y Ac 2d equation, z Or i6+4z-}-z* = 14Z, or z*— ioz=— 16, Whence z* — ioz-|-25=:25 — 16=9 by completing the square ; Andz — 5 = ^9=3, orz=3 + 5 = 8'; . Consequently ;?= 14— ^'— z= 14 — 4 — 8 = 2, and the numbers arc 2, 4, 8. 9. The sum (j) and ,llie product (/>) of any two num- bers being given j to find the sum of the squares, cubeS| biquadrates, &c. of those numbers. Let the two numbers be denoted by x and y ; Then will 1^"^-^^^ \ by the question, z Eat x-\-y\ zz:;v*4-2yy4-jr*=:j* by involutlony And A'-f-2Arj;-|-^* — zxyzzis^ — 2/ by subtraction. That is, x^-^-y^zzs^ — 2^:;:= sum of the squares. Again, QpADRAtiC EQUATIONS. T^6j Again, x^+y* Xx+yz=s*^2px* by Inultiplication, Or x^+xyxx+y+y^sis^'-'ijp. Or x^-^sf+y^zzf^-^zsf by substikatiiig *f for its equal' xyi>Cx+y ; And therefore x^+y^=is^'*-^3fp:=z sum of the cubes. In like manner, x^+y^ Xx+y=:s^ — 31/ x jby multiplication, Or x^+xyXx*+y*+y^=s^^3s*f, Or w*+/Xx* — 2/-fjF '*=j^-^3x'/ by substituting / x j*— 2^ for }ts equal xyXx*+y^ ; And «Dnscquently, x^+y^zzs^-^^s^p-^pxs^ — 2/=/* — 4^* /+2/*zz: sum of the biquadrates, or fourth powers ; and so on, for any power whatever. 10. The sum {a) and the sum of the squares (^) of four numbers in geometrical progression being given ; to find those numbers. Let X and y denote the two means, ^ , X* y* Then will -*— and — be th» two extremes, by the hature y X of proportion. ^ Also, let the sum of the two means nzj, and their product = p. And then will the sum of the two extremes =:a — s by the question, And their product 1= / by the nature of proportion. fx^ +y' =s'—2p -1 Hence < x^ y^ _ :* > by the last probleir^. Cx^ +y' =s'—2p -1 x^ V* ■" * And x*+y*+ -7 + - =rx* 4./1— j| —4^=^ by th6 quest- ion, y ** X* y* Again, f- •'^ zza — s by the question, J' * Or j<-:^ ^ |6| ALGEBRA. Or *p'4-jr'=:xjf X^f— / =/X «— ii But x*+j^'=rj*— 3x/ by the lastproblenif And therefore /xa— /=/'— 3j]^ by equalit/» Or/ia-^+3^=:/fl+2/x=:/% Or/=.^; tf+2X \Vheii(5e/*+ii— #|--4/=/* + ii-^l i^ =:* by siib- stitution^ • Or x' + — x=2 by reduction. And jrT'v/l— + ^ by coxapledng the square^ 2 4J* 2il and extracting the root. And from this value of / all the rest of the quantities p^ X and y may be readily determined. !^£STIOXS FOR PRACTICE. ^ 1. What two numbers are those, whose sum is 20, and their prouuct 36 ? Ans. 2 and 18. 2. To divide the number 60 into two such parts, that their product may be to the sum of their squares in the ratio of 2 to 5. Ans. 20 and 40. 3. The difference of two numbers is 3, and the diflfer* encc of their cubes is 117; what are those numbers ? Ans. 2 and 5;. 4. A company at a tavern had 81. 15s. to pay for their reckoning ; but, before the bill was settled, two of them sneaked otF, and then those, who remained, had los. a piece more to pay tlian before ; how many were there in the company ? , Ans. 7, 5. A 5* A grazier bought as many sheep as cost him 6oh and after reserving 15 out of the numbefi he sold the re« mainder for 54lt and gained as. a head by them ; how ma- ny sheep did he buy ? Ans. 75^ 6. There are two numbers, whose 4ifiecence is 15, and hsdf their product is equal to the cube of the less number ; what are those numbers ? Ans..3 ^^ ^^« 7. A person bought cloth for 33L ijfs. which he sold again at 2I. 8s. per piece, and gained by the bargain as much as one piece; cost him i required the number o£ pieces. ^ * Ans. 15. 8. What number is that, which being divided by the product of itp two digits, the quotient is 3 -; aiid if 18 be added to it, the digits will be inverted ? Ans. 24. 9. What two numbers are those, whose sum multiplied by the greater is equal to 77 ; and whose difference mul- tiplied by the less is equal to i2 ? Ans. 4 and 7. ID. When will the hour,' minute and second hands of a clock lie all together next after la o'clock ? *■ .Ans. Only at 12 o'clock. 11. The sum of two numbers is 8, and the sum of their cubes is 152 ; what are the numbers ? Ans. '3 and 5; 12. The sum of two numbers is 7, and. the sum of their fourth powers is 641 j what are the numbers ? ■ ' Ans. 2 and 5. 13. The sum of two numbers is 6^ and the sum. of their fifth powers is 1056 ; what are the numbers ? !fc ' Ans. *2 and 4. 14. The sum of four numbers in arithmetical .ptogrps- «ioa is 5^, and the sum of their squares is 464 ; wh^.are the numbers? .. Ans. jS^^xa, 16 and 20. .1 .'.:..': ;... • . .. ••;-: .- ! .: (.I^.-T* Y T ^'jd ALCEBHi. 15. To find four numbers in geometrical progression^ whose «um is i$i and the sum of tlicir squares 85* ' - Ansf ij 2, 4 and 8t ' i& Giren a?*— -rl +«*— -r = — ; to find -the x^\^ x\ a Takie ol If. CUBIC AND HIGHER EQUATIONS. A CUBIC EQUATION^ or equation of the third degree or- power, is one, that contains the third power of the un- known quantity : as x^ ^"^aX* -^-bxzzc. A biquafiratiCi or double quadratic, is an equation, that contains the fourth power of the unknown quantity : as ^* — ax^-^bx* — cxZ=Ld. An equation of the fifth powei^, or degree, is one, that contains the fifth power of the unknown quantity : as a^ -'^ax * '■\'bx * — ex *^dxzze. An equatioti of ths sixth power ^ or, degree, is one, that contains the sixth power of the unknown quantity : as a* •— jAT^-f"**'^* — ^^^ -H/^^ — ex-zif And so on, for all other higher powers. Where it is to be noted, however, that all the " powers, or terms, in the equation are supposed to be freed from surds, or fractional exponents. There are various particular rules for the resolution of cubic and higherequations ; but they may be all easily re- solved by the following rule of Double Position. '-" ' B.ULE, CUBIC AHD HIOHEI^ EQpAllONS. ^jt. ^ULE.* . t • Find by trial tMro numbersj as near the true root iu poa-» lible, and substitute them separately in the given equation,, instead of the unlcnown quantity ; rnarking the errprs> ;wlijich arise from each of them. 2. Multiply the difference of the two numbers, found by trial, by the least '^rror, and divide the prpduct by the difference of the errors, when they are alike, but by their fiiun, when they are unlike. Or say, as th^e difference or sum of the errors is to the difi^jence of the two numbers, $0 is the least error to the correction of its supposed number* 3. Add the quotient last found to the number belonging to the least crroi', when that number is too little, but sub- tract it, when ioo great j and the result will give the true lX)Ot nearly* 4. Take this root and the nearest of the two former,*^or my other, that may be found nearer ; and, by proceeding in like manner as above, a root will be had still nearer than before ; and so on, to any degree of exactness requir- ed. Each new operation commonly doubles the niunbe; of true figures in the toot. Note i. It Is. best to employ always two assumed num- bers, that shall differ from eagh other only }Ssj unity in the last figure on the right hand \ because then the differ- ence, or multiplier j| is only i. EXAMPLES. * This rule may be used fbr solving the questions of Double Position, as well as that given in the Arithnaetic^ and is prefera- ble for the present purpose. Its tryth is easily deduced from the tame supposition. For, by the supposltiod, r i s :i x — a : x — h therefore, by diWsiPHf r — X : J :: i— tf : x— i i which is the r^le. ftZAH^LBS. t; To fitid tbe it>ot of the cubic equation iv^-f-:r*-f-;ij trztoOf or tfle Value of at in it Here it is ^on foiind, that 9C lies between 4 and- 5. Assume, thereforet these two numbers, ^d the operatioa 1^1 be as fellows : tit iD])posinoii. ad sqipofttioiK 4 » 5 ' itf sf* . ay 64 «• ' 125 84 . Aums 155 —16 errors +55 The sum of which is 71, Thtti, as 71 : t : : i^ : •225, Hence Af=4'225 nearly. Again, suppose 4*2 and 4*3, and repeat die work af IpUows : I St suppositiqq. 2d sappositioiu 4*2 ^ 4'3 17-64 sc* 18-49 74-088 fC^ 79*5^07 u 95-928 sums 102*297 m^ -4*072 errors +2*297 The sum of which is &^6g. As 6-369 : -I : : 2*297 : 0*036 This taken from 4*300 Leaves ^ nearly z=i 4-264 CUBIC AND HIGHSE CQtTlTIOKS. }?} Ag«ii> sti^ose 4*264, and 4*265, and work ^ follows : Jit flappositiop. 4-2«4 18*18165^$ 77-526752 9^ z.^ supposition* 4265 1 8- 190225 77-581310 99-972448 sums 100-036535 . '02755a errors +0-036535 The sum of which is -064087. 'SThen, ^ '064087 : 'ooi : : 4*264 ; 0*0004299 To this adding 4*264 ^e have 9{ ycfy nearly =4*2644299 2. To find Ac toot of the equation (tf^-*'i5^*-}-63J^r: jfo, or the value df A^ in it. Here k aoon appears, that 9C is very little above x. {Suppose, thereforcj I'o and 1*1, and Urork as follows : VO $C ' ' VI 63-0 ' 49 63^ •m5«* #!une^ ^9-3 18*15 ^•33i 52-481 IP— I errors +2*481 3*481 sum of the errors. As 3*48 1 ; •! ? : 1 : •029 torrectJ i*6o lf83ro29^)earIf. Agaioi 374 ^JUbOMttA. ..-:» Aguni %\ippoic the tin> ^uttmbart V03 -md r^,i|nd vork as follows ; . . ro3 ro% * i'092727 • x^ 64*26' ' -1 5 '6066 l*OtJl20t>- *- 50-069227 sums * 49.7x520** I ■ " > " i> . . • ■ f i * -J--069227 crrora ' •— •2847^* •284792 As •354019 • "OX :: "069227 ^ This taken from 1*03 Leaves x nearly 1= 1*02804 'P ' l i> U ■ ■' Note 2. Every equation has as many roots as it coir4 talus dimensions^ or as there* are units in the index of its highest power. That.isi a simple equation has only pne value or root ; but a quadratic equation has two values or roots ; a cubic equation has tKree roots ; a biquadratic equation has four roots^ and so on. And when one of the root^ of an equation has" "been found by approximation, as before, the rest majr be found as follows :— Take for a dividend the given equatipni with the known term transposed, its sigh being changed, to the unknown side of -the equatipn } and for a divisor take a: minus the root just found. Divide the said., dividend by the divisor,, s^nd ^e quotient \rill be the equation depressed a degree lower than the given onel Find a root of this new equation by approximation, as before, and it- will be a sei^opd |r:oot of the original equa- tion. Then^ by meaa^- of this roct^ depress the second equation CtmiC AND ftieillBll-EQpATIONS. 37 J eqiia^ta rnicr degree lofwer ^ and thence' find a third root, tnd to on, till thie equation be reduced to a quadratic i then the two roots of this being found, by the method of completing ,the square, they will make up the remainder of the roots. Thus, in the foregoing equation, having found one jTOot to be i*o^8q4, connect it by minus with X for a divisor^ and take the given equation with the known term transposed for a divideojil : thus. Then the two roots of this quadratic equation, or X* — I3*97i96;tf=:-— 48-63627, by completing the square, are 6*57653 and 7'39543> which are also the other two roots of the given cubic equation. So that all the three roots of that equation, viz. x^— •i5x'-f-.63A:ZZ5o, aire 1*02804 and 6*576S3 . ^^^ . 7'39J43 Sum 15*^0006 And the sum of all the roots, is found to be 15, being ^aal to the .coefficient of the second term of the equation, which the sum of the roots always ought to be, when they are right. . ., . . Note 3. . It is also a particular advantage of the fore- going rule, that it is not necessary to prepare the equa- tion, as for other rules, by reducing it to the usual £nal form and state of equations.. Bedause the rule may 'be ap- plied^ enee-^an unreduced equation, though it be ever so much embarrassed by surd and compound^WHiUties. As in tho'foltewing xizTtiifit :'- ' :.•:;•• 3. Let it be required to find the root x of the equation ■"■' ' ^ 1 ^ ■— " ' ' ' _• . ' " 1 it *^ V^t44/r*.^/^^attf<f ;^ 114, or the Value of ^ in it. By By 91 &v trial! it is 00011 founds that die. rake of ll it W little ^X3ve 7* Suppose thectfore firsts that xzzj, and then that tszsS. Krstf when ic3:7. Second, trfien irrstf^ 47*906 v^i44**— jr^+2o|* 46*476 65*384 y/i96«*-i-«»+24l* 69-283 X 13*290 the sums of these 1 15*759 X 14-000 ^ the true number 1 14*000 >7io die two errors +i*759 As 2*469 : I i: cyio : o'i nearly. 7-0 HfSi7*2 nearly. Suppose again ^'tz^tf and^ because it turns out took great, suppose also ltf=7'i. Suppose Af=:7*2* Suppose *=7-i* 47-990 ^I44**'^-:^* +2o|* 47*973 66*402 v^i96»*— Af* + 24| 65*904 114*39^ the sums of these 113*677 114-000 die true number 114*000 -f-o'392 • the errors -t-^*ia3 0-123 ■■ * mm *5i5 : •123 :: T : '024. the correction* 7-100 . .. prh«cfctc.,x^7*i,24nearlytt^.ro9tr^^ CtJBIi; AND HIGHER EQUATIONS. 377 Note 4. The same rule also, among other more diffi-» tfcult forms of Equations, succeeds very well in what are icalled exponential equations^ or those, which have an un- known quantity for the exponent of the power 5 as in the following example. 4. To find the value of 9C in the exponential equation Jtf*Z=IOO. For the more easy resolution of this kind of equations. It is convenient to take the logarithms of tliem, and then compute the terms by means of a table of logarithms. Thus, the logarithms of the two sides of the present equation are, x^ log. of ^^2, the log. of 100. Then by a few trials it is soon perceived, that the vahic of x is soiliewhere between the two numbers 3 and 4, and indeed nearly in the middle between them, but rather nearer the latter than the former. By taking therefore first xzmzy^y and then xz=:y6i and working with the logarithms, the operation will be as follows : First, suppose flfzi 3*54 Logarithm of 3*5 = 0*5440680 Then 3*5 X log- 3'S = 1-904238 The true number 2*000000 Error, too little, Second, suppose X == Logarithm of 36 zz Then 3-6X log. y6 = The true number —•095762 3-tf- 0-5563025 2*002689 2'000000 Error, too great, +=■002689 -—'095762 + •002689 •09845 1 sum of the errors. Then, Z 2 As 378. * iLGEBRA. As *09845i : •! :: '002689 : o*66273. Which corrtction^ taken from 3*60000 Leaves 3*5972 7 =r;if nearly. On trial, this is found to be very little too small. Take therefore again a:= 3-59727, and next xz::2'S9T^^f and repeat the operation as follows : First, suppose *.=: 3*59727. Logarithm of 3*59727 is c'S55973i 3-59727 X log. of 3-59727= 1-9999854 The true number 2:0000000 Error, too little, ^— 0*0000146 Second, suppose yi;r3'59728. Logarithm of 3-59728 is o 5559743 3-59728 X log. of 3-59728= 1-9999953 ■ The true number 2*0000000 Error, too little, —-0-0000047 — 0-0000146 *— 0*0000047 0-0000099 difference of the errors. Then, As -0000099 : -ooooi :: '0000047 : 0*00000474747 Which correction, added to ' 3*59728000000 Gives nearly the value of x zz 3-59728474747 5. To find the value of x in the equation a: ^ -|- 1 cat * -{- 5^11:2600. Ans. ;vzz 1 1-00673- 6. To find the value of x in the equation X^ — 2Arz=5. Ans. 2*00455 n 7. To CUBIC AND HIGHER E<yJA'fI0N5. 379 7. To find the value of s in the equation ^^-f-2;c*— • %2X^=^lo. Ans. Ar=5'i349. 8. To find the value of ^ in the equation at'— I7flf*-|- 54VZZ350. Ans. /v=:i4-95407. 9. To find the value of 9C iu the equation /v*— -3^;*— 75^=10000. Ans. xzi:io'26i^, 10. To find the value of a* ih the equation 2^?*— -16:^^ ^740^' — -.30x1=:— r I. Ans. /v= I -284724. 11. To find the value of X in the equation Ar^-|-2;c*-j- 3A:3+4Ar*+5A;=5432i. Ans. ;vz=8-4i4455. 12. To find the value of x in the equation x^zz 123456789. V Ans. ^ii:8'64oo268. t^ T fF^ ^iSTD <>/• ALGEBRA. ^ / ^»o«<;>*o<»<s>»o<»o«<3*' geometry; DEFINITIONS, !• A. POINT is that, which has positioii| but not magnitude, 2. A line is length, without breadth or thickness. 3. A surface f or superficies, is an extension, or a figure, of two dimen- sions, length and breadth, but with- out thickness. 4. A tody, or solidf i$ a figure of three Tx" dtmeiisions, namely, length, breadth and thickness. ^, Hence surfaces are the extremities of solids ; lines the extremities of surfaces 5 and points the extremities of lines. 5. Lines * A Tutor teaches Simson^s Edition of Euclid's El^-. P^T? ^f Geometry in Harvard College, 361 GEOMETRY, 1 5. Lines arc cither right, or curved, or mixed of these two. 6. A right line, or straight liney lies all in the same direction between its extrem- iticS) and is the shortest distance between two points. 7. A curve continually changes its direction between its extreme points. 8. Lines are either parallel, obliqug, perpendicular, or tangential. 9. Parallel lines are always at the same distance, and never meet, though ever so far produced* 10. Oblique right lines change their distance, and would meet, if pro- duced, on the side of the Jeast distance. 1 1. One line is perpendicular to another, when it inclines not more on one side tlian on the other. 12. One line is tangential, or a tangent, to another, when it touches it without cutting, if both be pro- duced. tl. hi ftpiNlTlO^i 383 . ■ 13. An angle is t - jnclinntlonj tyx opening, of two lines, having different directions, and meeting in a point. 14. Atigles are right or oblique, acute or obttise* 15. Aright aftii I- til..", which is made by one line perpcndicub.r to an-^ other. Or when ci},; aiij^Us on each side are equal to one . . rlicr, tliey are right angles. 16. An obIiq!,'s .ui-k 'r v:\'t^ Urliicll is made by two obUqi.- ines, and is either less or greater ai a right ^angle. 1 7. Art acute angle Is less than a right angle. 18. An obtuse angle is greater than a right angle. 19. Superficies are either plane or curved. 20. A ))la?je superficm^ pr.a plane j\s that, witji which a right line may, every way, coincide. But if^not^ it .is curved. 21. Plane figures are bounded either by right linc^ or curves. ft 22. Plane figures, bounded by right lines,^ have .ftamea according to the number of* thejbr sides, OT^aiigtes.; for tliey have as many sides as angles ; the least number being three. ^3- A i 3^4 . OBOMETR* 23. A ligure tf three sides and angles is called a triaft* gle. And it receives particular denoKuoattons from the tc* lations of its sides and angles* 14. An equilateral triangle is thati whose three sides are equal. 2]^. An isosceles triangle is that, wliich has two side$ equal. 26. A scalene triangle is that, whose three sides are all unequal. 27. A right-angled triangle is that, Which has one right angle. 28. Other triangles are oblique-angled^ and are either ob- tuse-angled or acute-angled. 29. An BSIVlflTlOtn.. 3»f . ^. An ohusf'^ngkd triangk has wni obtuse angle. 30. An acutt'-angkd triangle has all its three angles acute. '3.1* A figure of four sides and angles is called a quad* ri£ngk^ or a quadrilateral. . ^2. j^^parallebgram is a quadrilaterali which has bodi pttr of its opposite sides paralleL And it ^akes the fol- lowing particular names* 3^3* A rectangle is a parallelogram, having all its angles right. 34* A square is an equilateral rectan- gle, having all its sides equal, and all its angles right. 35. A rhomboid is an oblique-angled parallelogram. rj 36. A rhombus is an equilateral rhom- boid, having all its sides equal, but its angles oblique. AAa 37. A ^S6 MoiiETKtl' "' ^7* A trapezium is a quadrilaterali which has not both pair of its opposite^ sidev parallcL 38. A trapezoid has -trdj one pair of /~ opposite sides parallel. / . 39, A diagofial is a right line, joining any two opposite angles of a quadrilateral* ^o. Plane figures, having more than four sides, 'aire, in general, called polygbns ; and they receive other particular' names, according to the number of their sides or angles. 41. A pentagon is a polygon of five sides 5 a bexagm has six sides ; a heptagcriy seven ; an octagon^ eight \ a noii' agon^ nine ; a decagon^ ten ; an undecagon^ eleven ; and a dodecagon^ twelve. 42. A regular polygon has all its sides and all its angles equal. — ^If they be not both equal, the polygon is ir- regular. 43. An equilateral triangle is also a regular figure of three sides, and the square is one of four ; the former be- ing also called a trigon^ and the latter a tetragon. Pentagon. Hexagon. Heptagon. Keptagon. Octagom Npna^iin^^ UndecagQXK pccagon. . Dodecagon. 44, A circle is a plane figure, bounded by a curve line, called the circumference^ which is every where equidistant from a certain point within, called the centre* Note. The circumference itself U often called a circle* 45. The irt •EOHBTtT* 45. The ra£us tit % circle it « right line, drawn from the centre to the circumference* 46. The £amiter of a circle b % right line, drawn dirojagh the centre, and terminating in the circumference •n both sides. 47. An arc of a circle is any part of the circumference. 48. A chord is a right line, joining / the extremities of an arc. • 49. A sfgment is any part of a /I circle, bounded by an arc and its ( chord. \ BEVINITIOMS. iP9 50. A ienucirde is half the^ cir- d^> QT a segment cut off by a 4i:« ^meter. L._A 51. A sector is any part of' a ^ircle^ bounded by an arc^ ^nd two radii^ drawn to its extremis ties. 5 2. A quadrant i or quarter of a circlei is a sector^ having a quar* ter of the circumference for its arc, and its two radii arc perpen- dicular to each other. 53. The bilghti or altituiey of ^ figure is a perpendicular let fall from an angle, or its vertex, to* the opposite side^ called the base. 54. In a right-angled triangle^ the side opposite to the right angle is called the hypotenuse; ahd;,the other, two the legsy or sides^ or sometimes the base ^xA perpendicular. 55. When an angle is. denoted by three letters, of which one standa at the angular point, and the other twt> on the two sides, that, which stands at the angular point, is read in the mddle. 3> C 56. The I 390 «BOMST&r. .-/X 56. The clr^rumfcrencc of every circle Is supposed to be divided into 360 equal parts, called degrees ; and each de- gree into 60 minutes, each minute into 60 seconds, -smd so on. Hence a semicircle contains x8o degreesi and a quad- rant 90 degrees. 57. The measure of a right- lined angle is an arc of any cir^ cle, contained between the two. / lines, which fo^m that angle, the [ angular point being the tfentrc \ \ and it is estimated by the num«* *'*• bet of dcjjrccs, contained in that arc. HcRce a right angle is an 2;iglc of 90 degrees. • 58. Identical Jigures are such,, as have all the sides and all the angles of one respectively equal to all the sides and all the angles of the otherj each to, each ; so that, if one figure were applied to, or laid upon, the other, all the sides of it would es^actly fall upon .'and cover all the sides of the other \ the two becoming coincident* 59. An angle in a segment \s that, which is contained by two lines, drawn from any point in the arc of the . segment to the ex* tremities of the arc. 60. A right-lined Jigure is inscribed in a eircle, or tie circle circumscribes: it, when all the angular points of the figure are in the circumference of the circle. 61. A right-lined figure circumscribes a cir~ elcf or the circle is inscribed in it, when all the sides of the figure touch the circumference •f the circle. 62. One ^2. Om rtghf-Unid figure is inscr'thed m an^ 'ethery or the latter circumscribes » the forvier^ when ail the angular points of the former are placed in the sides of the latter. 63. Similar figures are those, that have all tlie angles of ©ne equal to all the angles of the other, each to each, and the sides about the equal angles proportional. ^ ...■.....'"/•■ , ^64. The perimeter of a figure fa the ^ni of ail its sides^-. taken together. 65. A proposition iS something, • "vrhicli i« either proposed to be done, or to be demonstrated, andisu^ither a problem \»r a theorem. * 68* A problem is something proposed to be done. "* 67. A theorem is something proposed to be demonstrated. 68« A lemma is something, which is premised, or pre- viously demonstrated, in order to render what follows more easy« 69. A coroliaryis a consequent truth, gained inunediatei* ly from some preceding truth, or demonstration. 70. A scholium is a TemarLi or> obdervatior^, made upoE* something preceding it. PROBLEMS. *»*a •EOMBTRt. PROBLEMS PROBLEM !• Ttf £nnae a given Une A B into two e^tial parts. From the centxes A and B» with tuiy radios greater than half A B, describe arcs> cutting each other in m and n* Draw the line mcn^ and it will cut the given line into two equal parta in the middle point C. PROBLEM \U To divide a ghen Angle ABC into two equal parts* trom the centre B, with any radius^ de- scribe the arc A C. From A and C, with one and the same radius, describe arcs, in- tersecting in m. Draw the line B m^ and it will bisect the angle^ as required. ~ ^' |C B ^11 N xd,'""^ Note. By this operation the arc A C is bisected ; and in a similar manner may any ^iven arc of a circle be bisected. problem IPKOBLEMSt 3» i^ROBLEM III. jTi ^vide a right angle ABC into three equal fartin Prom the centre B, with any tadius, describe the arc AC. From the centre A, with the same radius» cross the zit AC in /i ; and with the centre C, and the same radius, cut the arc AC in m. Then through the points m and n draW" B m and B n, and they will trisect the angle, ^s required. j^aoBLEM IV* To draw a Line parallel to a given Line A B* Case u^-^When the parallel Line is to be at m given Disn tance C •Je O A •^.. • -• V' A — ^ — ""• m u C -.. 1> B From any two points m and n^ in the line AB, With a rad- ius equal to C, describe the arcs r an^ o. Draw CD to touch these arcs, without cut- ting them, and it will be the parallel required. Case ^•'^^When the parallel Line is to pass through a given Point C. From any point /», in the ^ line AB, with the radius mC, ^'^ \ % describe the arc C«. From the centre C, with ^^ same I radius, describe the Take the arc C« in the compasses, and apply it from m t9 Through C and r draw DE, the parallel required. Bfib NoTfi^ r. 3P4 GSOMBTRT. KoTE. In praxticc^psurallcl lines are moxe easily drawn with a Parallel Rule. ' PROBLEM V. To erect a PerpefuUciJar from a giveti Potnl A in a given Litie B C. Cass \.—When the Point is near the middUof the Line. On each side of the point A, take -h^^. any two equal distances lim^ An. From 4 the centred m and- n^ with any radius- greater than Am or A//, describe two arcs intersecting in r. Through A and r draw the line Ar, and it will be the g.^ perpendicular tequired. xu. jc V Cask ^.^^When the Point is nebr the end of the Line^ With the centre A, and any ^. ir radiusj describe the arc mns. From the point w, with the same radius, turn the compasses twice over on the arc, at n and /. A- gain, with the centres n and x, describe arcs intersecting in r. ^ ""*. '^ Then draw Ar, and it will be die perpendicular required* Another Method* From any point m, as a centre, with the radius or distance m A, describe an arc cutting the given line in n and A. Through n and m draw a right line cutting the arc in r; Lastly, draw" h ;•, and it will be the perpendicular re- quired. Another E TROBLEMS. *3W Another MetHod. From any plane scale of equal parts, set off Am equal to 4 parts. With centre A, and radius of three parts, describe an arc. And with centre m, and radius of 5 parts cross it at «. Draw An for 3 ^ the perpendicular required. Or any other numbers in the same proportion, as 3, 4^ c, will answer the same purpose. PROBLEM VI. From a given Point A, out of a given Line B C, to let fall m Perpendicular. Case u^^^Wien the p»nt ii nearly opposite the middle of the Line. < ■ - With the centre A, and any r.id- -^ ius, describe an arc cutting B C in I fn and «. With the centres m and ' *^ and the same,^ or any other rad- b j> ius, describe arcs intersecting in r. ^*^ Draw ADr for the perpendicular required. ■2L^i "JO, Cass 2*-— JFJ^o /^ Point is nearly opposite the end of the Line^ . K From A draw ^my lihe A m to A meet B C, in any point m. Bi- ' y sect A m at ;;, and with the cen- /* tre «, and radius- A« or /««, dc- z*^ scribe an arc, cutting BC in D. / Draw A D, the perpendicular re- 35 r quired. ^ ' \ t Another J9«« eSOMETRT. Another Method^ From B or any point in B (^ ts a centre, describe an arc through the poiot A. From any other centre m in B C, describe another arc through A, cutting the former arc' again in n. Through A and n draw the line A D n ; and A D will be the pexpendicular required. Note. Perpendiculars. may be more readily raised and let fall, in Practice^ by means of a square or other fit ini* strument. PROBLEM VII. 7i divuU a given Line A B into any proposed number af^ equal Parts. From A draw any line . A G at random, and from B draw 3 D parallel to it. On each of these lines, beginning at A and B, set off as many equal parts, of any length, as A B is to be divided into. Join the opposite points of division by the lines A 5, i 4, 2 3, &c. and they will divide A B as required. y>i PROBLEM FROBJLEMfti 497 PROBLEM, yill^ HfO £viii a pven Lint A B in- the same proportit^ as another Line C D is divided. C «i* From^ A draw any line AE equal to C D, and upoti it traha- fcr the divisions of die line C D. Join B E, tnd parallel tq It draw the lines i x, 2 2» 3 3^ &c. and they will divide A 9 a^ Tcquired. • IIROBLJEM IX. jlt a given Point A, ifi a given line A B, to maie an AngU eqt4al to a given Angle C. With the centre C, and any.« Tadiu$, describe an arc mn.r^ "With centre A, and the same radius, describe the arc rx.-^ Take the distance mn in the com- passes, and apply it from' r to /. Then a line, drawn through A ^ /, will make the angle A equal |o the angle C, as required. ** — C PROBLEM X, At a given P^nt A, in a given Line AB, to male an Angle <f any proposed, number of degrees. - . . • ' With the centre A, and radin^ equal to 60 degrees, taken from, a scale of chords, d^^nbc an arc cutting A,B in m. Then take in the compasses the pro- posed number of degrees from the same scale of chords, and . apply 39H: 8B0METRT« apply them from m to m Through- the point n draw A /?» a»d it will make th^ sip^lc A ^f tlic number q£ d<- gi'ccs proposed. • Or the angle ipay be nis^ovlth. any divided arc^ or in- strument, by" applying the centre . to . the point A, and its radius along A B \ then make a mack n at the^ proposed number of degrees, tjiirough M'hich dr^w the line. 'An^ ta before. ■ .r ...■■•:!. Note. Angles of more than 90 degrees' are usually taken off at twice. . PROBLEM XI. To mcamre a given Angle A» Describe the arc tnn with the chord of 60 degrees, as in the last Problem. Take thp arc m n in the compasses, and that extent, applied to ihe chords, will shew the degrees in the given angle.. PROBLEM XIU To fnd the Centre of a Clrtle. Draw any chord A B 5 and bisect it perpendicularly with C D, which will be a diameter. Bisect CD in the point 0/ \yhich will be the csn- tr<,\ PROBLCM »iLOBLEMS« 395> PROBLEM XIII. 7i de/criii' the Circumference of arjCircU through three^ given 'Points A,B,p. I FrOm the middle point B drAW chords to the other tXvo points- Bisect these choifds perpendicu- larly by lines meeting in o, which -^ will be the centre. Then from the centre, o, at the distance p At . or oB, or oC, describe the circle. ' : • Note. In the same mariner may the centre 6['MXzxt »f a circle be found. - ' * PROBLEM. XIV# Through a given Point A io draw a. Tangent to a given CircU* Case u-^JThen A is in the Circumference of the' Ctr elf. From the. given point A, £ draw Ao to the centre of the circle. Then through A draw • B C perpendicular to Ao, and it will be the tangent rc-j, , quired. 40a . dlOMBTRT; Case l- — Wben hisma rf the Circun^enct;. From the given poiiit A draw Ao to the centre^ which biaect in the point m. VTixh the centre m, and radius m A or tn 0| describe an arc, cut- ting the given cirde in' ly. Through the {)oint8 A and ft draw the tangent B C* PROBLEM XV; To find' a third Proportional to ttvo given Lines AB^ Ad A- A- Place the two given lineSi AB» AC, making any angle at A, and join BC In AB uke AD equal to AC, and draw DE par- allel to BC. So shall A£ be the third proportional to AB and ^ AC. That is, AB : AC :i AC : AE. -B PROBLEM ^VI. To find a fiourtif Proportional to three given Lines AB^ AC, AD. Place two of them, A B, AC, so as to make any angle at A, and join B C. Place AD on A B, and draw D E paraUel to BC. So shall A £ be the fourth proportional r&« quired. Thatis, AB : AC :: AD : AE- A- A- A- ^ PROBLEM HaOBLBHa. 4**v * • ■ ■ * Vo Jind a mean Proportional k^ween two ^ven Lines AB, BC. Join A B and B C in one A— straight line A C, and bisect ^ it in the point o. . With the centre o, and radius o A or" c Cy describe a semicircle. Eject the perpendicular Bt), " / and it will be the mean pro- j\. Ir- poj^ticnal require^ .. , . , •J^haris, AB : BD :: BD;: BC, -B O.JS PROBLEM XVIII* Ji divide a Line A B in Ex^treme and Mean RMio. Raise B C perpendicular to A B, 9nd equal to half A B. Join A C. With centre . C, and radiiis C B, cross AC in D. La'stly, with cen- tre A, and radius A D, cross * A B in E, which will divide the line A B, in extreme and n\can ratio, namc- \jy SO that the whole tine is to the greater part, as the greater part is^ to. the les9: part;^ That is, AB : AE :: AE : ETJ. Ccc PHOBt^EM !(et GMMtnT. PROBLBIf XIZ« To inseribi an isosedes triangle in a pven arcUf thai stfoB have each tf the angles at tie base double tie angle at the verten* Draw any diameter A B of the gives circle ; and divide the radius CB^ in the point Dy in extreme and mean ratio, bj the last problem. From the point B apply the chords BE, BF, each equal to CD; then join AE, AF, EF, and A £ F will be the triangle required. PROBLEM XX» To make an equilateral Triangle on a given Line AB. From the centres A and B, with the radius A B, describe arcs intersecting in C Draw A C and B Cj and it is done. Note. An isosceles triangle may be made in the same manner, by taking for the radius the given length of one of the equal sides. problem -^ Jr&OBLllii* PROBLEM XXU m Tq mah a Triangli vnti One ghen Lines AB, AC, B C. ' With the centre A and radius AC, describe an arc With the centre B and radios B C^ describe another arc cutting the former in C. Draw AC and BC» and ABC is the triangte required. PROBLEM JiXlU To mah a Sfimre i^m a pvm Lim AB. Draw BC perpendicuhir and equal to AB. From A and C^ with the radius ABj describe arcs intersecting in D. Draw A D and CD, and.it is done. Another Way* On the centres A and B^ with the radius A B, describe arcs crossing at a Bisect A o in n. With centre o, and rad- ius o/i| cross the two arcs in C and D. Then draw AQ BD, CD. PROBLEM An t^u^rM. PROBLEM XXIir. Si deicrtbe m Rcciangte^ or a ParalUlogram^ of m ^v^^ Length and Breadth. Place BC perpendicular to .^ A B. With centre A, and rad- ' ius D C, .describe au arc. With ce!itre C, and radius A B, de- scribe another arc cutting the A.*" former in D. Draw A D and C D, and it is done. B- 1 -C NoTi:. In tlie same manner is described any oblique parallelogram, except in drawing B C so as to make the given oblique angle with A B^ instead of* a right^uc PROBLEM XXIV. 2o tnnJ:e a regular Pentagon on a given Line A B. / •:\ .. rv ^Like B m perpendicular and equal to halt* AB. Draw A;;r, ,. ''^/ and produce it till m n be equal to B m* With centres A and B, and rail ius B /;, describe arcs intersecting in c, which will be the centre of the cir- cumscribing circle. Tlieu with A •J the centre c, and the same rad- ius, describe the circle 5 and about the circumference of it apply A B the proper r^uniber of times. . Another MLOBLfilM; Another Method. «P$ Make JB m peqiendicular and equal to A B» Bisect A B in /f ; then with the centre ft, and radius n m, cross A B produced in o. With die centres A and B, and radius Ao, describe arcs intersect- •ng in D, the opposite angle uf .the pentagon. Lastly, -- with centre D, and radius A B, cross those arcs ngdn in C and E, the other two angles of the figure. Then draw the lines from angle to angle^ to complete the figure. A third Method, nearly true. On the centres A and B, Inritb tlie radius A B, de- scribe two circles intersecting in m and If. With the same radius, and the centre m, de- scribe r Ao B S, an4 draw m n cut- ting it in o. Draw r o C and S o £» which will give two angles of the pentagon. Lastly, with radius A B, and centres'C and E, describe arcs intersecting in^D^ ^c other ai^gle of the pen- tagon nearly. - ' ■ _^ fROBLEM t 4o6 geomeYrt. PROBLEM X&V. 70 make a Hexagon on a given Lim A B« With the radius A B, and the centres A and B, de» scribe arcs intersecting in o. With the same radius, and centre o, describe a circle, which will circumscribe the hexagon. Then apply the line A B six times round the circumference^ marking out the angular points, which connect with right lines. PROBLEM XXVI. . -> To male an Octagon on a given Line A B. Erect A F and BE per* pendicular to A B» PrOi* duce A B both ways, and bisect the angles m A F and 19 B £ with the lines A H and B C, each equal to AB. Draw CD and H G parallel to A F or BE, and each equal to A B. With radius A B, and centres G and D, cross AF and BE, in F "^ ^ ^ and E. Then join G F, FE, ED, and it is done. •^J ^"\ A \ s d \ s P&OBLBIi FitOBLEM XXVII. To make any regular Polygon on a given Line A B. Draw A o and B oV mak- ing the angles A and B each equal to half the ai\gle of the polygon. Wicli the centre o, and radius o A, describe a circle. Then ap- ply the Hofi A B continually round the circumference the proper number of timest NoTB. The angle of any polygon, ojt >^?i;ch the angles o AB and oBA arc each iunz half, is fobi>u thus : divide the whole 360 degrees by the narnber of sides, and the quodent will be th<; angle ar ihe centre ; then subtract that from 180 degre^s^ and the remainder will be the an- gle of the polygon, fmd is ilonble of o A B, or of o B A. And thus you will find the following iablc, containing the degrees, in -the angle o jm ' rentrt td the .ngle of the polygon, for all the reju r. '..jures ■ ;ri 3 sides. No. of Nome of flir — Anr.'t: ■.■ jA the" An?'c of *l;e Uigko AB rid«8. Polygon. r-.ir.'e. r-iIyROn. 60° ar BA' '• ' 3 Trigon '.K** 30- 4 Tetragon 9- 90 45 5 Pentagon *?.; 1-8 54 6 Hexagor. -.- ,„•. 60 < ^ li Heptagci- _^ : , I -2 8-*. 64^ ■ 8 Octagor. 45 'l.-;':; ^>7t . 9 Nonage /^. ;^ 1-^,0 ^0 *" 10 Dccagc V •,"> 144 -2 II Undec -r-u '^^TT 1471V .3^ 12 Dodecagv > '^ 150 75 t^ PROBLEM 1 «ct QBOMETKT. f.ROBLEM XXVIII. /;; a given Circle to inscribe any e^iar polygon / #r, io dir*iiA tie Circumference into any nutfih^ cfi equal ParU^^ [Sec the last Fi^c.]. At the centre o make an angle equal to the angle stf tht centre of the polygon, as contained in the third coliimii of the above table of polygons* Then the distance AB vill be one side of the polygon ; which, 'being carried round the circumference the proper number of times, will complete the figure. Or, the arc A B, will he one of fhc cquil parts of the circumference; ■ ■' Another Method, nearly trae^ Draw the diameter AB which divide into as man^y equal parts as the figure has sides. With the radius A B, and centres A and B, describe arcs crossing at n ; whence draw n C through the second division on the di- ameter 5 so shall A C be a side ■of the polygon ncajly. ! yi Another \ t»ROBLEMS« 4of Another Method, still nearen Dividfe the diatrietet AB Inta ^ fis many equal parts aa the figure A has sides, as before. From the / \yj^ centre o raise the perpendicular oiw, which produce till m n he three^fourths of the radius o m. From n draw n C through the second division of. the diameteri and the line AC ^i\l be the side , of the polygon still nearer than before } or the arc A C, one pf * . ' the equal parts, into which the circumference is to be divided. PROJ&LEM ^XIX* AbotH a given Circlt to circumscribe any FalygoH^ Find the points f«, «, /, &c. as in th^* last problem, to which draw ri ii mo, no, &c. to th^ centre oJc tiic circle. Then through these ftcints i», », &c. and perpendicular to these radii, draw die sides of tbp. ,3C»lygon. Dpd tROBLEM <IO OEOMETiCT. PROBLEM XXX. To JhtJ the Centre of a given Pulygati^ or the Centre of its hi'- scribed or circumscribed Circle, Bisect any two sides with the per« pendiculafs* mo, no ; and their inter- section will be the centre. Then^ with the centre o, and the distance o/7/> de^ribe the inscribed circle \ oc •with the distance to one of the an- glesy as Aj describe the circumscribe ing circle. ^""^ Note. This method will also circumscribe a ciTcIcr about any given oblique triangle; PROBLEM XXXI. In any given Triangle to inscribe a Circle: Bisect any two of the angles with the lines Ac, Bo, and o will be the centre of the circle. Then, with the centre o, and radius the nearest distance to any one of the sides, describe the circle. PROBLEM XXXIII. Alout any given Triangle to circumscribe a Circle^ C Bisect any two of the sides A B, B C^ wirh the perpendiculars wo, no. Wirh the centre o, and distance to anyone of the angles, describe the circle. PROBLEM PltOBLEMS. 4if PROBLEM XXXIII. /«, or about y a given Square to describe a Circk. Draw the tM'o diagonals of the square, and their intersection o will be the cen- A tre of both the circles. Then, with that ^ / j centre, and the nearest distance to one I side for radius, describe the inner circle ; V, and with the distance to one angle for radius, describe the outer circle. PROBI.EM XXXIV, Irtjor ahoutya given Circle to describe a Square y or an Octagon, Draw two diameters A B, CD, perpendicular to each other. Then connect their extremities, and they will give the inscribed square A C, ^ B D. Also through thqir extremi- ties draw tangents parallel to them, and they will form the outer square mnop. It Note. If any quadrant, as A C, be bisected in <r, it will give one eighth of the circumference, or the side of the octagon. PROBlEM ^1*-' 1 4i» eSOMETRT. PROBLEM XXXV. In a given Circle to inscribe a Trigon, a Hexagon, or a Dodecagon, The radius is the side of the hexagon/' Therefore, from any point A in the circumferancc, with the distance of the radius, describe the arc BoF. Then Is A B the side of the hexagon ; and therefore, being carried round six times, it will form the hexa^^on, or divide the cir- cumference into six equal parts, each containing 60 degrees. The second of these, C* will give A C, the side of the trigon, or equilateral tri- angle, and the arc A C one third of the circumference, or- T20 degrees. Also the half of AB, or A r, is one twelfth of the circumference, or 30 degrees, awd give? the side of the dodecagon. Note. IF tnngents to the circle be drawn through all the angular points of any inscribed figure, they will form the ?idcs of a like circumscribing figure. PROBLEM XXXVK In a given Circle to inscribe a Pentagon^ or a Decagon, Draw i\\Q two diame- ters AP, rfin^ perpendic- ular to cacli other, and Lisect tlie radius en at ^. With llie centre ^,. and rr.divi q A, ('.cscilbe tl:c arc A r j aiul \\'J\ tlic cciiti'i A3 n:*ci iavili;r> Ar, desovJbi: tl;o r.'-c 1 B. Then lb A B one vS\\\ of the circumference } P^OBLBMS^ 413 circumference ; and A B, carried round five times, vill foim the pentagon. Also the arc A B, bisected in S, will give A S, the tenth part of the circumference, or the side 9i the decagon. Another MethocU Inscribe the isosceles triangle ABC, having each of the angles ABC, ACB, double the angle B A C- Then fais^cct the two arcs A D B, A E C,' in ^e points D, E \ and draw the chords AD, DB, AE, EC ; so shall ADB CE be the inscribed pentSgon re- quired. And th^ decagon is thence obtained as before. Note. Tangents, being drawn through the angular pointSj will form the circumscribing pentagon or decagon. PROBLEM XXXVII. fo divide the Circumference of a given Circle into twelve equal Partfy each being 30 Degrees. Or to inscribe a Dodecagon by another Method. Draw tv9to diameters i 7 and ^nd 4 10 perpendicular to each other. Then, with the radius of the circle, and the four extremi- ties 1, 4, 7, 10, as centres, de- scribe arcs through the centre of the circle ; and they will cut the circumference in the points re- quired, dividing it into .12 equal parts at the points marked with the number^. PROBLEM 414 CEOMETRY. P ROLL EM XXXVIII. 7o divide a given Circle inio any proposed Number of Parts by equal Lines ^ so that thse Parts shall be mu:ualh equals botk in Area and Perlm:ier. Divide the diameter AB into the proposed number of equal parts at the points a^ bf r. Sec. Thenon Afl, A^, Ar, g^c. asdian^- cters, describe semicircles on one side of the diameter A B ; and on B</, Br, Bby 8<c. describe semicircles on the other side sf the diLim'Jter. So shall the corresponding joining seniin circles divide the given circle in the manner proposed. And ill like m.inner we may proceed, when the spaces are to be in any given proportion. As to the perimeters^ they are always equal, whatever may be the proportion of the spaces. PROBLEM XXXIX. On a given Line A B /o describe the Segment of a Circle^ capable of coniaining a given Angle. Draw AC and BC, making the angles BAG and ABC each equal to the given a'lgle. Draw A D perpendicular to AC, and BD per- pendicular to B C. With centre D, and radius D A, or D B, de- scribe the se-mcnt AEB. Then any angle, as E, made in that seg- ment, will be equal to the given angle. PROBLEM t>R0BLEM3U ;4i5 PROBLEM XL. Si iut off a segment from a given Circle^ that shall cMairi s giveit Angle C. Draw any tan* gent A B to the given circle ; and a chord A D* to make the angle DAB equal to the given angle C ; then D E A will be tl\)e segment required, any angle E made in it being equal to the given angle C PROBLEM XLI* ^fi male a Triangle similar to a given Triangle A 1^ C. Let ab be/dne side of tJie re- quired triangle. Make the angle a equal to the angle A, and' the an- ^ gle h equal to the angle B ; then the triangle ah c will be similar to A B Cj as proposed. Note. If a i be equal to A B, the triangles will also be equal, as well as similar. PROdLEH 4i< OBOMETRT. ?ltOBLBM XLIU STtf tnaie a JPigun similar to any ether given Figure ABCDE. From any angle A draw (tiagonals to the other angles. Take hb a side of the figure required. Then draw be parallt^l to B C, and cdtQ CD, and de to D£j &c. li B j^ Otherwise. Make the angles at a^ b, e^ &c. respectively equal to the anghs at A»B»£j and the lines will intersect in the angles of the figure re- quired. PROBLEM XLIII. Ti make a Triangle equal to a given Trapezium A BCD. Draw the diagonal DB, and jy CE parallel to it, meeting AB produced in E. Join D E ; , so shall the triangle ADE be equal to the trapezium AB CD. ^ PROBLEM ^ mOBLEMS; «'?, To male a Triangle eqtutl to tbe Figure ABCDE AJ Draw the diagonals t) A, DB» md the lines EF, CG, parallel td them^ meeting the ba$e A^» both ways produced^ in F and G. JoinD^,D6) andDFQ lyiU be the triangle required* Note. Nearly in the sanie manner may a triangle W tnade equal to any right-lined figure whatever. >ROBLEM kLV* To maki A tticiahgie^ or a ParaHebgfath^ eqUdt ib a fivifi Triafigk A&Qi tAstct the hsat A B in 19^ Hirough C draw Cno parallel to AB. T^roUgh^ m and B draw mn and Bo parallel to each otiier, and either perpen* dicular to AB^ or making any angle with it. And the rectan- gle or parallelogram mffoB will be equal to the triangle^ as required. Efit PROBLEM 4/» C^tOMETHr. PROBLSM XLVI«r To male a Square equal to a given Rectangle ABCD. F,..^ Produce one side AB^ till B E be equal to the other side BC Bisect A E in o ; oiT which as a centre^ with ra4ius A Or describe a semicircle, add _ produce BC to meet it at F. '^, . .^ ^ On BF make the square BFGHf and itwiU be eqiial ttl the rectangle A B C D> as requir^. : - L iy- PROBLEM XLVII. . To maie a Square -equal to two given .SqHafeS ^ and Q* Set two sides AB, B G, of the given squares perpendicular to <ach other. Join their ex- tremities A C ; so shall the square R, constructed on A C, be equal to the two P and Q^ taken together. Note. Circles, or any other similar figures, ^re added in the sftme manner. For if A B and B C be the diame- ters of two circles, A C will be the diameter of a circle equal to the other two. And if AB and B C be the like sides of any two similar figuries, then A C will be the like side of another simiter figure equal to the two for- mer, and upon which the third figure may be constructed, Dv Problem xlii. PROBLEM PROBLEM XLVUIi? SJS To make a Square equal to tie Dlfereme of two gfveh Squares P, R. On the side A C of the greater square,' z» a diameter, describe a semicircle ; ia which apply AB the side of the less square. Join BC, and |t will be the side of a square equal to the difference- between ^he two P and R, as required. •PROBLEM XLIX» Xo tnttke 0. Square eqtiol U the sum of any nt/mher of Sqtiares tahn together. r Draw two indefinite lines Aw, A /7, ' perpendicular to each other, at the point A. On one of these set off AB. the side of one* of" the given squares, and on the other A C the side of another of tlxem. Join BC, and it will be the side of a square equal to the two together. Then take AD equal to BC, and A£ equal to the side of the third given square. So shall D E be the side of a square equal to the sum of the three give.i squares. And so on continual! js always setting more sides of the given squares on the line A ;;, and the sides of the successive sums on the other line A m, NOTF.. • • - - \ / / ^ ■ // /^ -^ . I> ]& ^ 1 42^ KomnY. NoTB. And thus any nuniber of any kud of tfmfH naj be added fq|;ether« rnoBLBii Is^ Ti9 cmsin^ th$ bus rf tU Pian^ Seak^ T^e dhriuons on tlie Plan^ Scs^e are of two kinds ; one kind having rebtion merely to right UpeSf and ^ other to the cirde and its properties. The former are called Linet^ or Scahs^ of Equal Parts^ and are either utnpli o^ diagonal. By tie Lines of the Plane Scale we here mean the follow-./ 4ng Line^ii most of which commonly, and att of ibxaao^ sometimes, are drawn 911 a PLiQe Scalf* |. A Line or Scale of Equal Parts^ marked E. K a « , • Chords, • • « • Cho,. 3**«»r*** Rhumbs « « • « ^hn* 4. •-••.• • Sioes t • « % % Sin, 5- •..•.. • Tangents n \ ^ ^ .Tan^ 6. • . • • • • Secants « « • • Sec« 7 • • • Semitangetits « • . £(• T*- 9 Longitude • • • « Lon. 9 Latitudes • • • . Lat. 10 Hours .... Ho# I ^ Inclmation of Meridians ' In. Men I. 5i I, Tif constryct plane diagonal Scaks^ 4fti Draw any line, as A B, of any convenient length. Di^ Tide it into 1 1 equal parts.* Complete these into rectan-i glcs of a convenient height, by drawing paftallel and per- pendicular lines. Divide the sdtitude into lo equal parts^ if it be for a decimal scale for common numbers, or inta I a equal parts, if it be for feet and inches \ and through these points of division draw as many parallel lines, the whole length of the scale. Then divide the length of the first division A C into lo equal parts, both above and be- low ; and connect tliese points of division by diagonal lines, and the scale is Jtmshedi after being numbered as you please. fLJNE SCALES jf on rnro Fjgu^s. >«'">'' '.I I ., do $ 6 4L 2 O ' ;c Luiiilmiil: ^ lO S 6 ^Z O f Only 4 parts arc here drawa for want of room. ^2% e^METRT. Pf the preceding three forms of scales for two figure* the first is a decimal scale, for taking off common num- bers consisting of two figures. The other two arc duo^ decimal scales, and serve for feet and inches. In order to construct the other lines, describe a circunv- feivincc with any convenient radius, and draw the diame- ters A B, D E, at right angles to each other ; continue BA at pleasure toward F ; through D draw DG parallel toBF; and draw the chords BD, BE, AD, AE. Cir- cumseribe the circle with the square HMN, whose sides, II M, MN, shall be paraUelto AB, ED, 2. To construct tie Line of Chords^ Divide the arc AD into 90 equal parts ; mark the loth cHvisions with the figures 10, 20, 30, 49, 50, 60, 70, 80, 90 \ on D, as a ccnirc, with the compasses, transfer the several divisions of the quadrantal arc, to the chord AD, which marked with the figures corresponding, will be a line of dioi-ds. Note. In tJie construction of this and the following scales, only the primary divisions are drawn ; the inter- mediate ones are omitted, tliat the figure may not appear toa much crowded. 3. TI; cchsti'iict the Line of Rhumbs. Divide the are BE into 8 equal parts, which mark with the figures 1, 2, 3, 4, 5, 6, 7, 8 j and divide each of those HoIbLemsV 413 J, n. nr. iv, v. vi. V 424 CIEOMETRT. those parts into quarters ; on B, as a ceAtre* teuisfer tbd divisions df the arc to the chord B E, which, marked with the corrcsponiling figures^ will be a line of rhumbs. 4. To cohjtridct the tine of Sihes.* Through each of the ^visions of the arc At) draw Tight lines parallel to the railius AC ) and CD will be divided into a line of sines, which are to be numbered from C to D for the right sines ; and from D to C fot the versed sines. The versed- sines may be continued to 180 dcgrcesyby laying the division^ of the radius CD from C toE. 5. To construct tie Llni of Tafigents* A rule on C, and the several divisions of the arc A D, will intersect the line D G, which will become a line of tangents, and i« to be figured from D to G with 10, 20| 30, 40, &c. 6. 7i construct tie Line of Secants* The distances from the centre C to the divisions on the line of tangents, being transferred to the line CF from the centre * For Definitions of Sides, Tatigents and Secants, seePLAK"! TaiooNOMETRY ; atid for that of Rhumbs, see Navigation. PROBLEMS. -425 <c:-.tre C, will give tbe divisions of the line of secants $ .vhich must be numbered from A toward F with tp^ 2o» y^ 7> cotutntct jliit Uttt rf SemitahgetitSp or the Tangents of half the Arcx. /. rule oti fif and the several divisions of the arc A D, viii mtersect the radius C A, in the divisions of the semi px half tangents ; mark these with the corresponding {^« ures of the arc A D. The semitangents on the plane scales are generally con^ tinned as far as the length of the rule, on which they are l^id, will admit \ the divisions beyond 90^ are found by dividing llie ate A£ like the arc ADj then laying. a rule by £ and these divisions of the arc A£, the divisions of the semitangents above 90 degrees will be obtained on the line C A continued* 8. To tonstruei the Line of Lonptude. Divide AH into 60 equal parts ; through each of these divisions parallels to the radius AC, will intersect the arc A£ in as many 'points ; from £> as a centre, the di- visions of the arc £A, being transferred to tlie chord £A, will give the divisions of the line of longitude. The points thus found on the quadrantal arc, taken from A to E, belong to the sines of the equally increas- ing sexagenary parts, pf the radius ; and those arcs, reck** oned FFf 47^ GEOMETRY. oned from E> belong to the cosines of those sexagenary parts. 9. 51? construct the Line of Latitudes* A rule on A, and the several divisions of the sines on CD, will intersect the arc BD, in as many points } on B, as a centre, transfer the intersections of the arc B D, to the right line B D ; number the divisions from B to D with, 'Oj 20j 30, 3cc. to 90 ', and BD will be- a Hnc of htitu^cs. 10. To co7istruct the Line of Hours* Bisect the quadrantal arcs B D, B E, in ^i, ^ ; divide {*:j quadrantal arc at into 6 equ^l parts, which gives 15 de- grees for each hour ; and each of these into 4 others, which wiH give the quarters. A rule on C, and the several divisions of the arc ab^ will intersect the line MN in the houri &c. points, wliich are to. be ma: ked as in the figure. 11. To construct the Line of Inclination of Meridians. Bisect the arc EA in c \ divide the quadrantal arc be in- to 90 equal parts ; lay a rule on C and the several divisions of the arc hcy and the intersections of the line H M will be tlie divieions of a line of inclination of meridians. -^ja^^ END OF VOL UME FIRST. I I r^ s.